On optimal currents of indistinguishable particles
OOn optimal currents of indistinguishable particles
Mattia Walschaers,
1, 2, 3, ∗ Andreas Buchleitner, and Mark Fannes Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Heverlee, Belgium Physikalisches Institut, Albert-Ludwigs-Universität Freiburg,Hermann-Herder-Str. 3, D-79104 Freiburg, Germany Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS,ENS-PSL Research University, Collège de France, 4 place Jussieu, F-75252 Paris, France (Dated: September 27, 2018)We establish a mathematically rigorous, general and quantitative framework to describe currentsof non- (or weakly) interacting, indistinguishable particles driven far from equilibrium. We derivetight upper and lower bounds for the achievable fermionic and bosonic steady state current, respec-tively, which can serve as benchmarks for special cases of interacting many-particle dynamics. Forfermionic currents, we identify a symmetry-induced enhancement mechanism in parameter regimeswhere the coupling between system and reservoirs is weak. This mechanism is broadly applicableprovided the inter-particle interaction strength is small as compared to typical exchange interactions. ∗ [email protected] a r X i v : . [ qu a n t - ph ] M a r I. INTRODUCTION
Currents—the specific physical feature of non-equilibrium steady states of open systems subject to a potentialgradient established by reservoirs—are a prominent topic of various branches of condensed matter physics. As thesizes of technological devices driven by currents reach mesoscopic scales, non-trivial quantum effects must be takeninto account [1–4]. Yet, our theoretical understanding of currents in quantum systems is far from complete, e.g., many results are available for perfect lattices [5], but more realistic set-ups, with disorder and decoherence, still posea panoply of open questions [6–12].The past decade has seen a vivid debate on the relevance of quantum mechanics in biological systems and mostnotably in photosynthesis [13–19]. Since photosynthetic organisms are immersed in an environment of thermal photons,one may describe the situation via a constant influx of photons triggering an outflow of electrons [20]. The system,a large collection of intricately coupled chlorophyll molecules, is therefore constantly experiencing a flow of excitonswhich may be interpreted as a current. At present, the debate [21–25] on how such flow in the stationary state canbe affected by quantum coherence on transient time scales remains widely open.Quantum effects do not only emerge naturally in many cases, they can also be engineered. In quantum dots [26–29]and molecular junctions [30–34], currents have been studied for decades. In addition, cold atom [35–38] and trappedion [39] set-ups provide clean testing grounds to study currents in a manifestly quantum mechanical setting, includingquantum many-particle and statistical effects.The aim of this contribution is to provide a rigorous mathematical treatment of currents in non-equilibrium quantumsystems. To achieve this goal, we need a model which is analytically controllable. Therefore, we treat the couplingbetween the system and the particle reservoirs in a Markovian way, i.e. we ignore memory effects in the dynamicsand use therefore a dynamical semi-group. Moreover, we focus on systems in which inter-particle interactions aresufficiently weak, such that the system can be described by effective model of free, i.e., non-interacting particles. Forfermions, this implies that the shifts in energy levels associated to inter-particle interactions must be small compare tothe energy-level spacings associated with the exchange interaction induced by the exclusion principle. In this scenario,we can derive bounds on the current, which are sufficiently tight to be saturated by properly designed systems. Ourresults thus also serve as a benchmark for studies of quantum transport in systems where interactions (or othernon-linear effects) cannot be ignored. A violation of our analytically derived bounds is an unambiguous indicator ofnon-trivial interaction-induced effects, beyond mere many-particle interferences between indistinguishable particles.To establish such a versatile theoretical approach which can handle the above diverse scenarios, and, in particular,also accounts for potential quantum statistical effects on transport, Section II of our present contribution providesa self-contained introduction to the mathematically rigorous framework of many-particle quantum currents. We willherein strongly rely on algebraic quantum statistical mechanics, a formalism which stems from mathematical physics.This algebraic approach to quantum mechanics of many-particle systems is indispensable to study infinitely largesystems (as we do in Section V). Within this framework, we introduce a quantum version of the continuity equation,applicable to open system dynamics of the semi-group type [40, 41]. We consider three contributions to the dynamics:a Hamiltonian part for the reversible particle dynamics, and two non-Hamiltonian parts which describe particleinjection and extraction, respectively. For such non-equilibrium many-particle systems we derive several fundamentalproperties: In Sections III and VI we derive an upper bound for the particle current in the fermionic setting, anda lower bound for bosonic systems, respectively. In Section IV, we show that the fermionic upper bound can besaturated by appropriate design of the Hamiltonian part of the dynamics. The algebraic framework allows us to gobeyond the standard Fock space formalism, which we illustrate in Section V, where we derive an upper bound for thecurrent density in a ribbon, i.e., a 2D lattice system with shift-invariance in one direction, and a finite width in theother.The strength of our contribution is that it makes no assumptions on the underlying single-particle Hamiltonian, andthat it is applicable whenever the interaction between particles can be ignored to a good approximation. Hence, ourapproach does not only provide fundamental insight on the achievable currents in non-equilibrium quantum systems,but also opens novel perspectives for research in the fields mentioned above, where one may exploit the here identifieddesign principle in a specific context.
II. MANY-FERMION SYSTEMS
We first provide an introduction to the algebraic formalism which describes many-fermion systems. The resultspresented in this section are well-known in the mathematical physics literature on quantum statistical mechanics [42–45]. In Sections III to VI, we apply this formalism to investigate the physics of currents in open quantum systems.
A. Fock space
It is common practice to describe many-fermion systems in terms of Fock space. This space is formally constructedusing a single-particle Hilbert space H , also referred to as the mode space, as basic building block which providesall degrees of freedom of a single particle. As postulated by Pauli, identical particles are independent of labelling, aconstraint which either leads to bosons or fermions. The wave functions of the latter species change sign under oddpermutations of particles which is reflected in the fermionic n -particle Hilbert space H ( n ) = H ⊗ H ⊗ · · · ⊗ H (cid:12)(cid:12)(cid:12) asym . (1)The anti-symmetrisation implies that the space H ( n ) is linearly generated by functions of the form ψ ∧ ψ ∧ · · · ∧ ψ n := 1 √ n ! (cid:88) π ∈ S n sign( π ) ψ π (1) ⊗ ψ π (2) ⊗ · · · ⊗ ψ π ( n ) . (2)Here S n denotes the permutation group of n objects, π a permutation, and sign( π ) the signature of π . Note that thesefunctions are generally not normalised and that they vanish whenever the single-particle wave functions are linearlydependent, as expected from fermions. Functions of the type (2) are often called Slater determinants.The fermionic Fock space Γ( H ) constructed on H is built to accommodate any number of particles and thereforeglues together all n -particle spaces: Γ( H ) = C ⊕ H ⊕ H (2) ⊕ H (3) ⊕ · · · , (3)where C describes the vacuum component where no particles are present in the system. In the fermionic case and fora finite dimensional mode space H the direct sum breaks off: Fermionic Fock space can never harbour more particlesthan dim( H ) . Often it is assumed that Fock space is sufficient to describe general many-particle systems which isslightly inaccurate. Fock space can only accommodate finite numbers of particles. Both in the case where H is notfinite dimensional, e.g., for infinitely extended systems (see Section V), or for a bosonic system (see Section VI),physics is much richer than Fock space. To study this larger realm of many-particle quantum physics, we must switchto a description in terms of observables and select a Hilbert space representation that matches the given physicalsituation. B. Algebra of observables
The main tools at hand in Fock space are the creation and annihilation operators: a † ( ϕ ) and a ( ϕ ) respectively,where ϕ ∈ H . We work in a formalism of non-local creation and annihilation operators which have a straightforwardinterpretation: they create and annihilate a single-particle state ϕ . Their action is easily given on Slater determinants: a † ( ϕ ) ψ ∧ ψ ∧ · · · ∧ ψ n = ϕ ∧ ψ ∧ ψ ∧ · · · ∧ ψ n , (4)where we have identified the n -particle vector ψ ( n ) = ψ ∧ ψ ∧ · · · ∧ ψ n ∈ H ( n ) with ⊕ ⊕ · · · ⊕ ψ ( n ) ⊕ · · · in Γ( H ) .The annihilation operator is the adjoint of the creation operator, its action on a Slater determinant is a ( ϕ ) ψ ∧ ψ ∧ · · · ∧ ψ n = n (cid:88) j =1 ( − j +1 (cid:104) ϕ, ψ j (cid:105) ψ ∧ · · · ∧ ψ j − ∧ ψ j +1 ∧ · · · ∧ ψ n . (5)Indeed, as one may expect for fermions, sign bookkeeping is required.Fermionic creation and annihilation operators obey the canonical anti-commutation relations (CAR) { a ( ϕ ) , a † ( ψ ) } = (cid:104) ϕ, ψ (cid:105) and { a ( ϕ ) , a ( ψ ) } = 0 ∀ ψ, ϕ ∈ H . (6)These operators generate an algebra that forms the basic mathematical framework for the description of many-fermionsystems with a given mode space H . The key idea of algebraic quantum physics is that the algebra of observables ,rather than a Hilbert space, is the central mathematical object to describe large quantum systems.As a general algebraic framework and to contrast it with the Fock space representation above, we introduce abstractcreation and annihilation operators c ∗ ( ψ ) and c ( ϕ ) respectively, ϕ, ψ ∈ H . It must be emphasised that these objectsare no longer linear operators on the Fock space, but merely generate a formal algebra determined by the basicrelations ψ ∈ H (cid:55)→ c ∗ ( ψ ) is C -linear (7) { c ( ϕ ) , c ∗ ( ψ ) } = (cid:104) ϕ, ψ (cid:105) and { c ( ϕ ) , c ( ψ ) } = 0 . (8)The ∗ is a formal operation which is the abstract version of the Hilbert space adjoint † . One then completes thealgebra with respect to the unique C ∗ -norm to obtain the C ∗ -algebra A CAR of the CAR on H .[46] The completionis needed to apply general mathematical results and to describe dynamics in a controlled way. This framework isnecessary to describe general many-particle systems with infinite-dimensional single-particle spaces; in these systems,we cannot describe all possible physics for all possible states (see Section II C) on the level of Fock spaces. In ourpresent contribution, we strictly require this framework for the study of the quantum ribbon in Section V.In this formalism, observables are those objects O ∈ A CAR which are constructed using , c ∗ ( ψ ) and c ( ϕ ) andwhich have the additional property that O = O ∗ . In the context of many-particle systems, it is often useful to focuson polynomials of c ∗ and c , in which contributions with specific particle numbers are related to definite orders. Inthis work, we focus solely on the simple class of single-particle observables corresponding to polynomials of order two.A single-particle observable is essentially an embedding of an observable on the one-particle space into the many-particle framework. In Fock space, one assigns a copy of the observable to each different particle in an additiveway, e.g., the total energy of a system described by a single-particle Hamiltonian is the sum of the single-particleoperators for each separate particle. The formal algebraic way to express this second quantisation is via the mapping Γ : B ( H ) → A CAR from the space of bounded operators on H to the algebra of observables, which acts as Γ( O ) := (cid:88) i,j (cid:104) η i , O η j (cid:105) c ∗ ( η i ) c ( η j ) (9)where we may select any orthonormal basis { η j } of H . In order to ensure that Γ( O ) belongs to the algebra one hasto impose the rather restrictive condition that O is a trace-class operator. It is not hard to check that different bases { η j } yield a same second quantised observable.A specific example of interest in the discussion of particle currents for finite dimensional one-particle spaces H isthe number operator N which literally counts the number of particles in the system. This operator is in essence ofsingle-particle type, as it is given by N := Γ( ) = (cid:88) j c ∗ ( η j ) c ( η j ) . (10)Indeed, particle currents describe the in- and outflow of particles and therefore the behaviour of the observable N goes hand in hand with the behaviour of such currents.Not only does the algebraic formalism require a more abstract description of the observables in our theory, it alsoimplies a more general structure for the quantum states which determine the statistics of measurement outcomes forthese observables. C. States
A quantum state is commonly associated either with a state vector ψ or with a density matrix ρ . Expected valuesof observables O are given by (cid:104) O (cid:105) = (cid:104) ψ, O ψ (cid:105) or (cid:104) O (cid:105) = Tr ρ O . This presupposes a specific Hilbert space representationof the physical system. The more general algebraic formalism starts with expectation functionals that allow for aprobabilistic interpretation [47–49]. Thus a state is a linear functional ω : A CAR → C on the algebra of observablesfulfilling the requirements ω ( ) = 1 and ω ( x ∗ x ) (cid:62) ∀ x ∈ A CAR . (11)These properties are respectively known as the normalisation and positivity conditions.A useful tool to describe states, and their perturbations, on a C*-algebra is the Gelfand-Naimark-Segal (GNS)construction [43, 45, 50, 51]. This procedure associates a unique, but state-dependent, Hilbert space representationof the algebra to state ω . This representation returns the state as an expectation with respect to a state vector.Different states may, however, lead to inequivalent representations, which typically happens in the thermodynamiclimit of many particle-systems. As an example one may consider Bardeen-Cooper-Schrieffer theory [52–59], wherestates with a finite particle density in the thermodynamic limit must be represented in a different Hilbert space thanthe Fock space which is constructed by exciting the physical vacuum (see Section II A). The GNS construction is a keyresult in algebraic quantum physics, which stresses that the properties of the system’s state are essential prerequisitesto study physical models.States on A CAR are usually characterised in terms of correlation functions , i.e., one strives to define all objects ofthe form ω (cid:16) c ∗ ( ψ ) · · · c ∗ ( ψ m ) c ( ϕ n ) · · · c ( ϕ ) (cid:17) . (12)In the present context, where only single-particle observables are considered, there are simple ways to describe therelevant expectation values. A notable fact is the existence of a linear operator Q ∈ B ( H ) for each ω , which serves asa (non-normalised) density matrix and is commonly interpreted as a covariance matrix: ω (cid:0) c ∗ ( ψ ) c ( ϕ ) (cid:1) = (cid:104) ϕ, Q ψ (cid:105) . (13)In the class of gauge-invariant quasi-free states, this operator Q suffices to fully determine the state.[60] In general,this is far from true and one can just say that Q characterises the single-particle statistics.The fact that we are considering states on the CAR-algebra directly implies that (cid:54) Q (cid:54) . The first inequality isnecessary to fulfil positivity of the state, the second is a consequence of the fermionic behaviour and represents Pauli’sexclusion principle. It follows [45] that for a general single-particle observable Γ( B ) ∈ A CAR , with B a trace-classoperator on H , ω (cid:0) Γ( B ) (cid:1) = Tr( Q B ) . (14)This identity might not seem spectacular but it offers an enormous computational simplification. It is, therefore, oneof the key ingredients in all the following sections of the present contribution.If ω is a normal[61] gauge-invariant quasi-free state it can be shown that ω ( N ) = Tr Q < ∞ , (15)hence directly expressing the expected particle number in terms of Q which is now also a trace-class operator. Thiscondition is also sufficient to guarantee normality. D. Dynamics
In the spirit of the algebraic approach, it makes sense to consider the elements of the algebra A CAR as the dynamicalobjects in the theory, whereas the states remain unchanged at all times. This formally implies that we can considera mapping Λ t ,t : A CAR → A
CAR for an evolution from time t up to t . The first obvious requirements for awell-defined dynamics are Λ t ,t ( ) = and Λ t ,t ( x ∗ x ) (cid:62) ∀ x ∈ A CAR . (16)These demands must be fulfilled for any choice of t and t .A more debatable [62–64] assumption on the dynamics is complete positivity which formally says that the systemcan be trivially embedded in a larger system without having to fear for loss of positivity. Such embeddings are alsoimportant to include internal degrees of freedom in the description. Complete positivity in other words guaranteesthat effective descriptions of only a subset of relevant degrees of freedom are possible. The formal mathematicalphrasing requires an extension of the algebra by any matrix algebra M N to obtain A CAR ⊗ M N . We may nowtrivially extend Λ t ,t on A CAR to Λ t ,t ⊗ id N on A CAR ⊗ M N . When Λ t ,t ⊗ id N is a positive map for any N , thedynamics is said to be completely positive [65–67].In addition to complete positivity, one may impose another demand which rarely holds exactly for a real physicalsystem but often provides a very good approximation [68–70]: We impose a one-parameter semi-group structure onour dynamical map. The term “semi-group” implies divisibility of the map and hence the existence of a generator .Moreover, the generator is time-independent and thus the map is only governed by t = t − t . In other words, wecan write the dynamics in terms of Λ t , and obtain that Λ t + s = Λ t ◦ Λ s = Λ s ◦ Λ t ∀ t, s (cid:62) . (17)In general, we do not assume that the inverse exists, thus withholding the family of maps from being a full-blowngroup.This type of dynamics is particularly useful due to powerful mathematical results. The results by Gorini, Kos-sakowski, Sudarshan [71] and Lindblad [67] are well-know, but only hold for algebras of observables which can bedescribed by bounded operators on a Hilbert space. Nevertheless, Lindblad provided a more general recipe for com-pletely positive, one-parameter semi-group dynamics on a C*-algebra A : He showed that any equation of motion ofthe type dd t x = Ψ( x ) + k x + x k ∗ ∀ x ∈ A , with k ∈ A and Ψ :
A → A a completely positive map , (18)leads to a dynamical map with such properties. Hence, we may follow this prescription to engineer a dynamical systemwith the desired phenomenological properties. In other words, we do not microscopically derive a master equationbut rather study one which has the correct phenomenology.In our present work, we follow and explore a model described by Davies [72]. From here onward, we assume that H is finite dimensional which is a considerable technical simplification. In section V, however, we will deal withtranslation-invariant systems and discuss how to cope with this more general situation. In particular, (18) allows usto write the generator of the dynamical semi-group in a form that nearly resembles the standard Lindblad form [72]: dd t x = i [ h , x ] + D d ( x ) + D a ( x ) , (19) D d ( x ) := (cid:88) j γ dj (cid:18) c ∗ ( δ j ) θ ( x ) c ( δ j ) − { c ∗ ( δ j ) c ( δ j ) , x } (cid:19) (20) D a ( x ) := (cid:88) j γ aj (cid:18) c ( α j ) θ ( x ) c ∗ ( α j ) − { c ( α j ) c ∗ ( α j ) , x } (cid:19) , (21)where α j , δ j ∈ H . To make sure that Ψ in (18) is a CP-map, one must impose [72] that θ in (19) is the ∗ -automorphismdetermined by θ (cid:0) c ∗ ( ψ ) (cid:1) = − c ∗ ( ψ ) , ψ ∈ H . (22)The Lindblad generators D a and D d describe the injection and extraction of particles into and from the system,respectively. With respect to the system degree of freedom, these terms mediate absorption and dissipation, thus thesuperscripts a and d . More specifically, fermions described by single-particle state vectors { α j } are injected into thesystem with positive rates { γ aj } , and particles which state vectors { δ j } are lost from the system with positive rates { γ dj } . Note that also temperature dependences can be accommodated within the positive rates { γ a/dj } . We considersystems of non-interacting particles, therefore, in accordance to (9), we must set h = Γ( H ) with H = H † ∈ B ( H ) . We follow the model of [72], and many results of that paper are relevant for the present one. Specifically, we areinterested in the dynamics of single-particle observables, given by x = Γ( B ) with B ∈ B ( H ) . Using (9), we insert Γ( B ) into (19) followed by a straightforward computation [69, 72, 73] based on the anti-commutation relations (8),and we find that the relevant equation of motion for one-particle observables is given by dd t Γ( B ) = i [Γ( H ) , Γ( B )] − { Γ( P ) , Γ( B ) } + 2 Tr (cid:0) AB (cid:1) , with P = A + D , A := n a (cid:88) j =1 γ aj | α j (cid:105)(cid:104) α j | , and D := n d (cid:88) k =1 γ dk | δ k (cid:105)(cid:104) δ k | . (23)That P, A, D ∈ B ( H ) directly follows from their definitions. Because the semi-group dynamics generated by (23) isMarkovian, all rates must be positive, which in turn implies that A (cid:62) , D (cid:62) , and hence P (cid:62) . Moreover, forconvenience in Section III B, we assume that A and D are strictly positive. For the bound on the current that will bederived later on, we can always consider the general case A (cid:62) and D (cid:62) using continuity. E. Non-equilibrium steady states
Now that the equations of motion are determined, we observe that they can be solved exactly: Λ t (cid:0) Γ( B ) (cid:1) = Γ (cid:16) e − ( P − iH ) t Be − ( P + iH ) t (cid:17) + 2 (cid:90) t d s Tr (cid:16) e − ( P − iH ) s Be − ( P + iH ) s A (cid:17) . (24)We notice that, through its dependence on the absorption generator A , the second term is specifically related to thepopulation of the system, via the particles that are pumped in. To infer the statistical distribution of measurementresults associated with the observable Λ t (Γ( B )) , we need to lift (24) to the level of states, by virtue of (14): ω (cid:16) Λ t (cid:0) Γ( B ) (cid:1)(cid:17) = Tr (cid:16) e − ( P − iH ) t B e − ( P + iH ) t Q (cid:17) + 2 (cid:90) t d s Tr (cid:16) e − ( P − iH ) s B e − ( P + iH ) s A (cid:17) . (25)An alternative perspective can be formulated by considering the object ω ◦ Λ t ; because Λ t describes a dynamicalmap, it actually follows that, for any t > , ω ◦ Λ t is a quantum state in its own right. In other words, we can treatthe dynamics in the Schrödinger picture, by defining a family of states ω t := ω ◦ Λ t . (26)Intriguingly, equation (25) even provides us with an explicit expression for the Q ( t ) that appears in (14); by rewrit-ing (25), we find ω t (cid:0) Γ( B ) (cid:1) = Tr (cid:0) BQ ( t ) (cid:1) , with Q ( t ) = e − ( P + iH ) t Q e − ( P − iH ) t + 2 (cid:90) t d s e − ( P + iH ) s A e − ( P − iH ) s , (27)where Q is the single-particle covariance matrix for the initial state ω. Typically, at asymptotic times, pumped systems relax into a non-equilibrium steady state where finite currents areflowing. This limiting state has completely forgotten the initial conditions of the system. Put differently, genericallyeach system observable converges to a multiple of the identity. The way to describe the asymptotic state, is byexplicitly considering the limit t → ∞ in (24). To do so, note that since P > , generically, lim t →∞ e − ( P + iH ) t Q e − ( P − iH ) t = 0 (28)and therefore we find that ω NESS (cid:0) Γ( B ) (cid:1) = Tr (cid:0) BQ NESS (cid:1) , with Q NESS := lim t →∞ Q ( t ) = 2 (cid:90) ∞ d s e − ( P + iH ) s A e − ( P − iH ) s . (29)Here NESS stands for non-equilibrium steady state. This, e.g., implies that the expected number of particles in thesystem converges to n = ω NESS ( N ) = 2 (cid:90) ∞ d s Tr (cid:16) e − ( P + iH ) s A e − ( P − iH ) s (cid:17) . (30)It is not hard to show [72] that the NESS state is the gauge-invariant quasi-free state determined by Q NESS . III. CURRENTS
Non-equilibrium systems are typically characterised by the presence of currents, even when they reach a stationarystate. In this section, we first discuss general properties of currents, determined by the “continuity equation” (33). Wetranslate these results to a quantum mechanical setting to arrive at a sound definition of quantum particle currents .Finally, we extensively discuss a fundamental property of fermionic currents, which is one of our key results:
Theexistence of a universal upper bound —irrespective of the specific potential encountered by the particle flow.
A. Particle Currents
We start by a formal definition of the particle current in the context of quantum master equations. The generalstructure in (19) presents us with the change of particles in the system over time. Because the number operator N = Γ( ) , we insert B = into (23) and obtain dd t N = D d ( N ) + D a ( N ) , (31) = (23) − D ) − A ) + 2 Tr (cid:0) A (cid:1) . (32)Note that the Hamiltonian contribution vanishes in the evaluation of (23) because [Γ( H ) , Γ( )] = Γ (cid:0) [ H, ] (cid:1) = 0 . Thisimplies that the Hamiltonian, which is itself an observable of the form (9), conserves the total number of particles.We now study the particle current as a thermodynamic flux [40, 41] and focus on its behaviour in the NESS (29).We note that, by definition of the steady state, the time derivative of the number operator is zero in the NESS, whichyields, after combination of (31, 32) with (14), the balance equation ω NESS (cid:16) D d ( N ) + D a ( N ) (cid:17) = (29) − (cid:0) DQ NESS (cid:1) + 2 Tr (cid:0) A ( − Q NESS ) (cid:1) = 0 . (33)We can now define the current of out flowing particles as J := (cid:12)(cid:12) ω NESS (cid:0) D d ( N ) (cid:1)(cid:12)(cid:12) = 2 Tr (cid:0) DQ NESS (cid:1) = (33) (cid:0) A ( − Q NESS ) (cid:1) , (34)where the absolute value is added because we focus on the magnitude of the current. B. Bounding the current
Although expression (34) suggests that the current is independent of the Hamiltonian H ∈ B ( H ) , it is implicitlypresent in Q NESS . Indeed, we can rewrite the current, using (29), to obtain J = 4 (cid:90) ∞ d s Tr (cid:16) De ( − iH − P ) s Ae ( iH − P ) s (cid:1) . (35)In principle, this expression allows for a direct computation of the current, although this is generically intricate, e.g., when the operators in (35) do not commute. It is therefore instructive to derive a bound, to gain some generalunderstanding of the parameter dependence of the current.To do so, we first introduce the super-operator G ( X ) := − i [ H, X ] + { P, X } with P = A + D . (36) G can be split into a sum of two commuting terms, left multiplication by P − iH , and right multiplication by P + iH ,respectively. Therefore, we may write G = L P − iH + R P + iH (37)where L Y ( X ) := Y X and R Y ( X ) := XY . (38)Generically, G is invertible and for positive definite P > we can use the identity G − ( X ) = (cid:90) ∞ d s exp( − s G )( X )= (cid:90) ∞ d s exp( − s L P − iH ) ◦ exp( − s R P + iH )( X )= (cid:90) ∞ d s e − s ( P − iH ) X e − s ( P + iH ) . (39)Next, we compute G ( X † X ) − G ( X † ) X − X † G ( X ) = − X † P X ≤ , (40)from which it follows that G ( X † X ) ≤ G ( X † ) X + X † G ( X ) . (41)We now introduce a symmetrised zero temperature Duhamel (or Bogoliubov) inner product [45, 74–76]: (cid:104) X, Y (cid:105) ∼ := Tr (cid:16) X † G − ( Y ) + G − ( X † ) Y (cid:17) . (42)Here X and Y are general linear operators. Positivity of the scalar product follows from the invertibility of G , from G ( X † ) = (cid:0) G ( X ) (cid:1) † , from (41) and from Tr G ( X † X ) = Tr { P, X † X } ≥ . (43)For an explicit evaluation of Schwarz’ inequality |(cid:104) A, P (cid:105) ∼ | (cid:54) (cid:104) A, A (cid:105) ∼ (cid:104) P, P (cid:105) ∼ (44)we observe that L P ± iH ( ) + R P ∓ iH ( ) = 2 P . (45)from which we infer that (cid:104)
A, P (cid:105) ∼ = Tr A, (46) (cid:104) P, P (cid:105) ∼ = Tr P, (47) (cid:104) A, A (cid:105) ∼ = 2 (cid:90) ∞ d s Tr (cid:16) A e − s ( P − iH ) A e − s ( P + iH ) (cid:17) . (48)Inserting these results in (44), we obtain (cid:0) Tr( A ) (cid:1) (cid:54) (cid:0) Tr( A ) − J/ (cid:1) Tr( A + D ) (49)and it then follows that J (cid:54) A ) Tr( D )Tr( A + D ) =: J max , (50)which is the desired bound to the current. It is a universal one, since it does only depend on the reservoir couplingagents A and D , but not on the potential landscape which the fermions have to be transmitted through, defined bythe system Hamiltonian H .Since J max lacks a dependence on the single-particle Hamiltonian H , it is suggestive to inspect the tightness of thebound (50) for variable relative strength of unitary dynamics and reservoir couplings. For this purpose, we slightlyrewrite (35) as J = 4 (cid:90) ∞ d s Tr (cid:16) D e ( − iλH − P ) s A e ( iλH − P ) s (cid:1) , (51)where we introduced the parameter λ , to scale the relative strength of the Hamiltonian part of the dynamics ascompared to particle loss and pump. λ → completely cancels the coherent part of the dynamics while λ → ∞ makesthe oscillatory Hamiltonian part much faster than the rates with which the systems couples to the reservoirs.In the remainder of this section, we seek to numerically confirm the validity of (50) when λ in (51) is varied.To approach this problem, we consider many realisations of the system, each time choosing a random λ , randomHamiltonian H , and random channels A and D . For every realisation, the NESS current (51) is evaluated andcompared to the upper bound (50). The results of this procedure are shown in Fig. 1, where specific choices for therandom matrix ensembles were made: We consider a system of m modes, i.e., dim( H ) = m . The Hamiltonian H is sampled from the Gaussian orthogonal ensemble (GOE) [77] which implies that it is a matrix whose entries aresampled from a normal distribution: H ij ∼ Normal (cid:16) , (1 + δ ij ) v √ m (cid:17) . (52)0 ������� ������� ● ������ ���������� - � �� - � ���� � ��� �� � �� � ������������������������ λ � / � � � � Figure 1. Scatter plot of the stationary current J (51) relative to the maximal current J max (50). The variable λ (51) controlsthe relative strength of the Hamiltonian and incoherent contributions. For each data point log λ is randomly chosen fromthe interval [ − , . For each realisation, the Hamiltonian H in (51) is chosen from the GOE (52) with typical coupling v/ √ m between modes, with v = 1 , and mode number m = 10 . The channels (23) A and D in (51) are drawn from a Wishartensemble (53) with m A = 5 and m D = 10 . Data points are compared to the upper bound J = J max (horizontal red line). Thevalue λ = 1 is indicated by a vertical grey line, it coincides with the mean eigenvalue of P = A + D . The parameter v is related to the spectral radius ( i.e., the largest eigenvalue in absolute value) and physically v/ √ m can be thought of as the typical ( i.e., root mean squared) coupling strength between different modes. The matriceswhich describe the couplings between the system and the channels must be constructed so that they are alwayspositive semi-definite. A standard method to generate random matrices fulfilling this constraint is to resort to theWishart ensemble [78]. The latter ensemble is solely determined by the number of absorption (dissipation) channels, m A ( m D ). For our numerical simulations, we set A = 1 m A + m D W † a W a , and D = 1 m A + m D W † d W d (53)where W a and W d are m A × m and m D × m matrices respectively. They are generated by choosing random componentsaccording to (cid:0) W a,d (cid:1) ij ∼ Normal(0 , . (54)The additional factor ( m A + m D ) − in (53) is included to set the average eigenvalue of P (23) equal to . Withthis choice of ensembles, we can genuinely interpret λ (51) as the ratio of the frequencies of the coherent oscillationsinduced by H and the incoherent rates contained in P. Fig. 1 clearly shows that the bound (50) is valid for all realisations regardless of the magnitude λ . Nevertheless, wedo observe that the bound is typically more accurate in the limit of dominantly coherent dynamics, characterised by λ (cid:29) . In this contribution, we will not attempt to understand the specific statistical properties which are obtainedfrom the random matrix theory treatment. We do, however, note that in the limit λ (cid:29) (or mathematically λ → ∞ ),the current is susceptible to changes in the Hamiltonian (recall eq. (51)) and that, therefore, a natural next step is toattempt to saturate the bound (50) in this regime.[79]1 IV. SYMMETRY ENHANCED CURRENT
In this section, we investigate how an appropriate design of the system can generate a current close to J max (50).Because we are considering a designed system, it is reasonable to focus on the regime λ (cid:29) where the coherentdynamics has a strong influence on the current. To get a maximal effect of the coherent contributions, we rigorouslyfocus on the regime λ → ∞ . This allows us to treat the problem using perturbation theory. In this limit, rapidlyoscillating terms appear in (51) and by the Riemann-Lebesgue lemma [80] many contributions to J cancel.The Hamiltonian can be represented in its spectral decomposition as H = (cid:88) k E k R k . (55)Here the E k are the eigenvalues of H and R k are the orthogonal projectors on the corresponding eigenspaces of H .Using first order perturbation theory (where /λ is small), we compute lim λ →∞ J = lim λ →∞ (cid:90) ∞ d s Tr (cid:110) De − ( P + iλH ) s Ae − ( P − iλH ) s (cid:111) = (cid:88) k (cid:90) ∞ d s Tr (cid:110) DR k e − sR k P R k Ae − sR k P R k R k (cid:111) . (56)If we want to saturate the bound on the current we have to design the R k in an appropriate way.Structuring Hamiltonians goes hand in hand with introducing symmetries. We therefore assume the existence of aunitary operator U , such that [ H, U ] = 0 . (57)In order for such a symmetry to be useful, it must connect the couplings of the absorption channels A to those of theoutput channels D , leading to the requirement U † A U = D . (58)Given these additional structures, we can rewrite (56) as lim λ →∞ J = (cid:88) k (cid:90) ∞ d s Tr (cid:110) e − sR k P R k R k U † A U R k e − sR k P R k A (cid:111) . (59)The fact that U and H commute, implies that U is block-diagonal with respect to the spectral decomposition of HU = (cid:77) k U k . (60)This further implies lim λ →∞ J = (cid:88) k (cid:90) ∞ d s Tr (cid:110) e − sR k P R k U † k R k A R k U k e − sR k P R k A (cid:111) , (61)which can in general not be cast in a more transparent form. However, in the case where the Hamiltonian H isnon-degenerate (implying that, apart from (57), there are no unitary symmetries present in the system), we obtainthat R k are rank-one operators . In this case we can express U as U = (cid:88) k e iθ k R k , (62)such that e iθ k are the eigenvalues of U . In turn this leads to lim λ →∞ J = (cid:88) k (cid:90) ∞ d s Tr (cid:110) e − sR k P R k R k A R k e − sR k P R k A (cid:111) . (63)By virtue of (56), where D is replaced by A , and (29), the right-hand side is exactly lim λ →∞ AQ NESS . Due to thesymmetry (57), this implies that Tr AQ NESS = Tr DQ NESS in the regime λ → ∞ . (64)2However, from the balance equation (33), we read that Tr DQ NESS = Tr A − Tr AQ NESS . (65)Both equations (64) and (65) can hold simultaneously only when Tr AQ NESS = Tr A/ , which implies that lim λ →∞ J = Tr A = Tr D . (66)The second equality in (66) follows from (58). Inserting expression (58) into (50) directly yields J max = Tr A = ⇒ lim λ →∞ J = J max (67)which is exactly what we wanted to achieve.In words, we have shown that, in the absence of degeneracies in H , it suffices to find a unitary operator U whichcommutes with the Hamiltonian ( i.e., a symmetry) and transforms D into A, in order to saturate the upper boundfor the current in the limit λ → ∞ . The most natural picture to associate with such a mathematical formulation, isthat of a reflection symmetry. The limiting regime of λ can be seen as a rigorous way of demanding weak coupling,implying that the time scales of the system dynamics are much faster than those set by the rates with which thesystem couples to its reservoirs.In realistic set-ups, this limit is never exactly achieved, therefore it is instructive to conduct numerical simulationsto assess the deviations from the optimally achievable current, as a function of λ . This is done in Fig. 2: To make thesimulation as general as possible, we start by sampling the unitary U introduced in (57) from the Haar measure [77].[81]Because the matrix is unitary and random, we can always obtain a spectral decomposition U = m (cid:88) k =1 e iθ k | e k (cid:105)(cid:104) e k | . (68)We use this decomposition as the starting point for the construction of the Hamiltonian, which we define as H = m (cid:88) k =1 E k | e k (cid:105)(cid:104) e k | E k ∼ Uniform (cid:0) [ − m/ , m/ (cid:1) . (69)So [ H, U ] = 0 follows by construction. The choice of the uniform distribution for the eigenvalues E k is arbitrary, itsimply serves to ensure that the typical level spacing—and hence the typical frequency of the coherent oscillations—isindependent of the system size N . We sample A from the Wishart ensemble, see (53) and (54), but must take theconstraint D = U † AU into account. When focusing on the regime of λ (cid:28) in Fig. 2, we observe a similar trend asfor Fig. 1. However, once we approach λ (cid:29) , we observe that, indeed, J ≈ J max for all realisations.The fact that the bound (50) can be saturated in the regime of dominantly coherent dynamics, λ (cid:29) , can beunderstood in a straightforward way: On the one hand, the rates with which the reservoirs couple to the system settime scales for particle exchange, which also governs the bound (50). On the other hand, however, coherent timescales, set by the Hamiltonian, determine how the particles explore the various modes inside the system. Therefore,if these coherent time scales are too slow, particles will linger in the modes where they entered the system where theyblock the path for additional particles due to Pauli’s exclusion principle. Hence, the limit λ (cid:29) , guarantees fastredistribution of particles within the system, and, in addition, the design principle (58) and (57) guarantees a balancebetween input and output channels, such that particles can be extracted efficiently. In general, however, interferenceeffects, incorporated in the fact that A, D, and H do not commute, make this naive picture more complicated. Thisis precisely why a mathematically rigorous treatment is important and non-trivial.Furthermore, we stress that this discussion makes statements on the current J (51) relative to the bound J max (50).However, the maximal current J max itself depends on the absorption and dissipation channels (as governed by op-erators A and D ). Therefore, when we rescale these parameters as A → γA and D → γD , it directly follows, fromexpression (50), that J max → γJ max . The results of Fig. 2 imply that for any such value of γ , the current J (51)can be optimised, for λ (cid:29) γ , by appropriately designing the system according to (58) and (57). However, becausethe value of the bound increases with γ , it is conceivable that a large value of γ (and thus large rates of particleexchange between system and reservoirs) can lead to large currents, even for slow coherent time scales, i.e., λ < γ .This hypothesis is, indeed, confirmed in Fig. 3. We can define the current J γ which results from rescaling A → γA and D → γD : J γ = 4 (cid:90) ∞ d s Tr (cid:16) γD e ( − iH − γP ) s γA e ( iH − γP ) s (cid:1) . (70)3 ������� ������� ● ������ ���������� - � �� - � ���� � ��� �� � �� � ������������������ λ � / � � � � Figure 2. Scatter plot of the stationary current J (51) relative to the maximal current J max (50). The variable λ (51) controlsthe relative strength of the Hamiltonian and incoherent contributions. For each data point log λ is randomly chosen fromthe interval [ − , . Each realisation of the Hamiltonian H in (51) is generated according to (69) with mode number m = 10 and random symmetry operator U from the Haar measure [77]. For each realisation, the absorption operator A (23) in (51) isdrawn from a Wishart ensemble (53) with m A = 10 . The dissipation operator D is determined by the condition (58). Datapoints are compared to the upper bound J = J max (horizontal red line). The value λ = 1 , which coincides with the meaneigenvalue of P , is indicated by a vertical grey line. Fig. 3 shows how the current (in units which are fixed by the Hamiltonian’s mean-level spacing) scales as a functionof the rescaling parameter γ . The Hamiltonians are generated following (52) and (69), for fully random and designedHamiltonians, respectively. For the fully random systems, a single set of input and output channels, represented by A and D , respectively, is randomly chosen according to (53) and kept fixed. In the simulation of the designed systemswe fix only A and generate D according to (58).In Fig. 3 we see that typically the current increases when we increase the incoherent rates for particle exchange(by varying γ ). However, we also observe that the bound is tighter in the regime of dominantly coherent dynamics asgiven by γ (cid:28) . In this regime, the designed systems give rise to optimal transport by saturating the bound, whereaswe see fluctuations in the full random systems (see inset in Fig. 3). It must be noted that the double logarithmicscale of the plot masks these fluctuation.Up to this point, we studied systems which contain a finite number of particles at all times. The strength of theC*-algebraic treatment and the formulation of the model in terms of a general CAR is the possibility to extend thesetting to systems with an infinite number of degrees of freedom. In the following section, we consider systems thatrequire a technical treatment based on current densities. We prove a generalisation of the bound (50) to a class ofshift-invariant systems as commonly encountered in theoretical solid-state physics. V. THE QUANTUM RIBBON
Above we focused on systems with a finite dimensional mode space, which excludes models with a translationalinvariance in some spatial directions. The latter situation requires to perform a thermodynamic limit, i.e., we firsthave to consider a finite subsystem, and subsequently perform a limiting procedure where the size of the systemtends to infinity while the particle density remains finite [45]. We now consider such a model situation, with someinspiration from [82].The specific system under consideration is a ribbon: A 2D system with translation invariance in one direction, andfinite width in the orthogonal dimension. We assume that the system is accurately described by a tight-binding modeland therefore the single-particle Hilbert space is given by a discrete lattice H := l ( Z ) ⊗ C d , where d quantifies thefinite width of the lattice. For k ∈ Z we denote by { k } the sequence in l ( Z ) with 1 at place k and 0 everywhere4 Bound ● Random Hamiltonians10 - - - γ J × - × - γ J J J Bound ● Random Hamiltonians10 - - - - γ J γ J J J Fully Random Systems
Designed Systems
Figure 3. Scatter plot of the stationary current J γ (70) in units of the mean-level spacing of the Hamiltonian. The variable γ (70) determines the incoherent time scales. For each data point log γ is randomly chosen from the interval [ − , . Datapoints are compared to the upper bound J max (red line, 50). Both fully random and designed systems are shown. In eachcase, an inset shows the fluctuations of the currents J γ as compared to the bound, by zooming in on the parameter range γ ∈ [0 . , . . Hamiltonians and channels are generated as follows: Fully random systems (top):
For each realisation, the Hamiltonian H in (70) is chosen from the GOE (52) with typicalcoupling v/ √ m between modes, with v = 1 , and mode number m = 10 . A single set of channels (23) A and D in (70) aredrawn from a Wishart ensemble (53) with m A = 5 and m D = 10 , and remain fixed for all realisations. Designed systems (bottom):
Each realisation of the Hamiltonian H in (70) is generated according to (69) with modenumber m = 10 and random symmetry operator U from the Haar measure [77]. A single absorption operator A (23) in (51)is drawn from a Wishart ensemble (53) with m A = 10 , which is kept fixed for all realisations. The dissipation operator D isdetermined by the condition (58). else. The mode space of our system can then also be seen as (cid:76) k ∈ Z { k } ⊗ C d with the one-step shift along the ribbongiven by { k } ⊗ ψ (cid:55)→ { k +1 } ⊗ ψ. A. Shift-Invariance
Let us first focus on the space l ( Z ) . The Fourier transform F : l ( Z ) → L (cid:0) [0 , π ) (cid:1) can be defined through itsaction on the indicator functions { k } ∈ l ( Z ) : F { k } := ϕ k , with ϕ k ( x ) = e ikx . (71)5A bounded operator operator A on l ( Z ) is shift-invariant if and only if A = F − M ˆ a F . Here ( M ˆ a ψ )( x ) = ˆ a ( x ) ψ ( x ) , ψ ∈ L (cid:0) [0 , π ) (cid:1) (72)and ˆ a ∈ L ∞ (cid:0) [0 , π ) (cid:1) . Therefore a bounded shift-invariant operator on l ( Z ) corresponds to a multiplication operatoron L (cid:0) [0 , π ) (cid:1) by a bounded function on [0 , π ) . Hermitian operators correspond hereby to real-valued functions andpositive semi-definite operators to non-negative functions.This can straightforwardly be generalised to H = l ( Z ) ⊗ C d : We say that a bounded operator X on l ( Z ) ⊗ C d is translation-invariant along the ribbon iff X = ( F − ⊗ ) M ˆ X ( F ⊗ ) , where ˆ X : [0 , π ) → M d is a boundedmatrix-valued function. If we now denote e k,l = { k } ⊗ e l , with { e l } the standard basis in C d , we may write that (cid:104) e k (cid:48) ,l (cid:48) , X e k,l (cid:105) = 12 π (cid:90) π d x (cid:68) e l (cid:48) , ˆ X ( x ) e l (cid:69) e − i ( k (cid:48) − k ) x . (73)It also follows that, for two shift-invariant operators X and Y,XY = ( F − ⊗ ) M ˆ X ˆ Y ( F ⊗ ) . (74)It is useful to generalise (73) to the case where ˆ X : [0 , π ) → M d is integrable.[83] In general, such a choice leads toan unbounded X .To discuss currents, we are confronted with the problem that the global number operator N , which counts thenumber of fermions on the ribbon, is not an element in the CAR algebra over l ( Z ) ⊗ C d . In fact, A does notcontain any shift-invariant elements except for the multiples of . Shift-invariant elements are introduced by theirlocal restrictions on finite subsets Λ ⊂ Z of the ribbon. To construct these local restriction, we define the appropriateprojectors P Λ := (cid:88) p ∈ Λ (cid:12)(cid:12) { p } (cid:11)(cid:10) { p } (cid:12)(cid:12) ⊗ . (75)We can now consider the Λ -restriction Γ( P Λ XP Λ ) of “ Γ( X ) ”, which is a bona fide element of the algebra. Translation-invariance manifests itself by Γ( P Λ+1 XP Λ+1 ) being the one-step shift of Γ( P Λ XP Λ ) . The global number operatorcorresponds to the choice ˆ X ( x ) = 1 for x ∈ [0 , π ) and its restriction to Λ is just the number operator for the modespace l (Λ) ⊗ C d , i.e., it counts the number of fermions on the compact domain defined by the restriction Λ .Suppose that a shift-invariant (cid:54) Q (cid:54) determines the one-particle expectations (13) and that X defines ashift-invariant one-particle observable as above. Both Q and X are determined by matrix-valued functions ˆ Q and ˆ X on [0 , π ) that satisfy the requirements that (cid:54) ˆ Q (cid:54) and ˆ X be real-valued and integrable. We can now considerthe expectation of the density of Γ( X ) in a specific state ω Q where we rewrite henceforth P n := P { ,...,n } ˜ x ( ω Q ) = lim n →∞ n ω Q (cid:0) Γ( P n XP n ) (cid:1) , (76)where we introduce the “ ∼ ” to refer to densities in the system. Because of translation-invariance, there is no problemin fixing the leftmost site of the interval at 1. A small computation, similar to the type of computations used inproving Szegö’s theorem [82], yields ˜ x ( ω Q ) = 12 π (cid:90) π d x Tr C d (cid:0) ˆ X ( x ) ˆ Q ( x ) (cid:1) . (77) B. Currents in the Quantum Ribbon
The bound (50) on the fermionic current nicely fits with shift-invariance. For shift-invariant H , A , and D, bothsides of the bound scale linearly with the length of the sub-interval on the ribbon that we consider. It then sufficesto renormalise the inequality to obtain an analogous bound for densities.The dynamics is a priori similar to the dynamics generated by (23), although we now specifically focus on thesituation where H, A, and D are shift-invariant operators. We can simply repeat the arguments from Section II Eabove and obtain that in the long-time limit any state asymptotically converges to a shift-invariant state determinedby the matrix-valued function x (cid:55)→ ˆ Q NESS ( x ) = 2 (cid:90) ∞ d s e − ( ˆ P ( x )+ i ˆ H ( x )) s ˆ A ( x ) e − ( ˆ P ( x ) − i ˆ H ( x )) s . (78)6The particle density ρ ( ω ) for a translation-invariant state determined by ˆ Q (77) is given by ˜ ρ ( ω ) = 12 π (cid:90) π d x d Tr C d (cid:8) ˆ Q ( x ) (cid:9) . (79)Note that in our dynamical system we are typically dealing with a particle density ˜ ρ ( ω ) that changes over time.We start by considering the evolution of the local number operator N n = Γ( P n ) , which is described by (23): dd t Γ( P n ) = i Γ (cid:16) [ H, P n ] − { A + D, P n } (cid:17) + 2 Tr (cid:0) P n AP n (cid:1) (80)such that dd t ω (cid:0) Γ( P n ) (cid:1) = i Tr([
H, P n ] Q ) − Tr( { A + D, P n } Q ) + 2 Tr (cid:0) P n AP n (cid:1) = Tr( P n [ Q, H ] P n ) − Tr( P n { Q, A + D } P n ) + 2 Tr (cid:0) P n AP n (cid:1) . (81)Note that, because P n is a projector, (14) yields ω (cid:0) Γ( P n ) (cid:1) = Tr (cid:0) P n QP n (cid:1) , and that (74) implies that commutatorsand anti-commutators of shift-invariant operators are again shift-invariant. Therefore, we may use (77) and evaluate dd t ˜ ρ ( ω ) = lim n →∞ n d dd t Tr (cid:0) P n QP n (cid:1) = 12 π d (cid:90) π d x (cid:16) Tr C d (cid:8) ˆ A ( x )( − ˆ Q ( x )) (cid:9) − Tr C d (cid:8) ˆ D ( x ) ˆ Q ( x ) (cid:9)(cid:17) (82)where we already used that Tr C d ( i [ ˆ Q ( x ) , ˆ H ( x )]) = 0 for all x. By definition of the non-equilibrium steady state dd t ˜ ρ ( ω NESS ) = 0 (83)and we therefore define the current density as ˜ j = 1 π d (cid:90) π d x Tr C d (cid:8) ˆ D ( x ) ˆ Q ( x ) (cid:9) = 1 π d (cid:90) π d x Tr C d (cid:8) ˆ A ( x )( − ˆ Q ( x )) (cid:9) . (84)For every x ∈ [0 , π ) , we can apply (44) to find (cid:0) Tr C d ˆ A ( x ) (cid:1) (cid:54) (cid:16) Tr C d ˆ A ( x ) − Tr C d (cid:8) ˆ A ( x )( − ˆ Q ( x )) (cid:9)(cid:17) Tr C d (cid:8) ˆ A ( x ) + ˆ D ( x ) (cid:9) (85)which can be rewritten and integrated to obtain ˜ j (cid:54) π d (cid:90) π d x Tr C d ˆ A ( x )Tr C d ˆ D ( x )Tr C d ˆ A ( x ) + Tr C d ˆ D ( x ) , (86)as a universal upper bound for the fermionic current across the quantum ribbon. VI. BOSONIC SYSTEMS
Throughout the preceding parts of this contribution, we focused on systems of non-interacting fermions. Ourmethods are, however, also applicable to systems of non-interacting bosons. In this scenario, we must consideradditional technical details related to the algebra of canonical commutation relations (CCR) [45, 51, 84]. One technicalissue is that, for infinite dimensional mode spaces H , the bosonic algebra only allows us to define creation andannihilation operators in a representation dependent way. Another technical issue is that, even for finite dimensional H , states are not necessarily given by a density matrix on Fock space. Therefore, we here deliberately focus on systemswith a finite number of particles, such that we remain in the Fock representation at all times.The bosonic Fock space is defined (quite analogous to (1, 3)) as Γ b ( H ) := C ⊕ H ⊕ H (2) ⊕ H (3) ⊕ · · · (87)7with H ( n ) = H ⊗ H ⊗ · · · ⊗ H (cid:12)(cid:12)(cid:12) sym . (88)The CCR can now we written in terms of non-local creation and annihilation operators b † ( ϕ ) and b ( ϕ ) which acton “Slater permanents” in a similar fashion as (4) and (5). These unbounded operators on Γ b ( H ) satisfy canonicalcommutation relations: ψ ∈ H (cid:55)→ b † ( ψ ) is C -linear (89) [ b ( ϕ ) , b † ( ψ )] = (cid:104) ϕ, ψ (cid:105) and [ b ( ϕ ) , b ( ψ )] = 0 . (90)In analogy to the fermionic case, we describe our dynamics in terms of the phenomenological master equation[85] dd t X = D ( X ) = − i [Γ( H ) , X ] + D a ( X ) + D d ( X ) , for all X ∈ Lin (cid:0) Γ( H ) (cid:1) , with D a/d ( X ) = (cid:88) i L a/di † XL a/di − { L a/di † L a/di , X } , (91)and, in analogy to (19-21) we choose L di = (cid:112) γ di b ( δ i ) and L ai = (cid:112) γ ai b † ( α i ) . Again, { α i ∈ H} denote the single-particlestate vectors in which particles are absorbed into the system, whereas { δ i ∈ H} denote the single-particle state vectorsfrom which particles are dissipated out of the system. Because the creation and annihilation operators are unbounded,it remains to verify that this leads to a valid dynamical map, i.e., that it fulfils the conditions (16) and maps elementsof the algebra onto other elements of the alegbra. To do so, we evaluate D ( b † ( ϕ ) b ( ψ )) = b † (cid:0) ( iH − P ) ϕ (cid:1) b (cid:0) ψ (cid:1) + b † (cid:0) ϕ (cid:1) b (cid:0) ( iH − P ) ψ (cid:1) + 2 (cid:104) ϕ, Aψ (cid:105) , (92)where the bosonic P is defined as P := D − A , (93)with A and D as in (23). A fundamental difference between the fermionic and bosonic case is that the bosonic P in (93) is not necessarily a positive semi-definite operator on the single-particle space.We use (92) to evaluate the dynamics of a general single-particle observable Γ b ( B ) = (cid:88) i,j (cid:104) η i , Bη j (cid:105) b † ( η i ) b ( η j ) , (94)where { η i } forms an orthonormal basis of the single-particle Hilbert space H . Straightforward integration of (91) leadsto Λ t (Γ b ( B )) = Γ b (cid:16) e t ( iH − P ) Be t ( − iH − P ) (cid:17) + Tr (cid:18)(cid:90) t d s Ae ( iH − P ) s Be ( − iH − P ) s (cid:19) . (95)We now observe that the case where P is not positive semi-definite can lead to severe problems because it typicallydoes not allow for the system to remain contained within the Fock representation at all times. This can be understoodby assessing the time evolution of the particle number expectation value. We consider systems which are initiallylocal with respect to the Fock representation, therefore the state is given by a density matrix ρ which acts on Γ b ( H ) and (cid:104) N (cid:105) ρ = Tr { ρ Γ b ( ) } < ∞ . (96)However, when P is not positive semi-definite, for generic ρ the asymptotic particle number is given by lim t →∞ Tr { ρ Λ t (Γ b ( )) } = ∞ . (97)and thus diverges in the long-time limit. Physically this means that the system is unstable and never reaches a steadystate. Therefore, we must impose that D (cid:62) A , (98)8and therefore P (cid:62) , to ensure that the system remain confined to Fock space for all all times. This implies thatsystems of non-interacting bosons which absorb particles from an external reservoir require a sufficient (as quantifiedby (98)) amount of dissipation to ensure the existence of a well-defined NESS.[86]Having imposed condition (98), we find that the solutions to the bosonic and fermionic equations of motion, (95)and (24), respectively, are very similar, such that the same analysis as above can be repeated. The bosonic continuityequation is the same as the fermionic one when we write it in terms of P : In the NESS we find P Q
NESS = 2Tr
A . (99)However, the definition of P has changed, so that the following balance equation between incoming and outflowingcurrents holds: (cid:16) A ( + Q NESS ) (cid:17) = 2Tr ( DQ NESS ) . (100)This implies that we can still describe the current flowing through the system as J := 2Tr ( DQ NESS ) . Remarkably, this definition of the current, together with (98), implies that we can next precisely follow the steps(36-48) of the proof for the fermionic bound. However, in (49) we did employ the explicit form of the continuityequation, and therefore this step differs from the present bosonic case. We now find that (Tr A ) (cid:54) ( J/ − Tr A )Tr P , (101)which implies Tr A (Tr A + Tr P ) (cid:54) J P (102)and therefore J (cid:62) A Tr D Tr( D − A ) =: J min . (103)The inequality (103) is remarkable because its derivation is largely analogous to that for the fermionic case, but itultimately produces a very different phenomenology: There is no upper bound for bosonic currents in the NESS.However, bosonic currents are always stronger than a given quantity J min which is set by the channels. In systemswhere A comes close to D , while respecting (98), we see that the rate at which particles stream through the systemcan become arbitrarily large.Finally, we numerically scrutinize the lower bound (103). These results shown in Fig. 4 are obtained throughevaluation of the exact expression for J : J = 4 (cid:90) ∞ d s Tr (cid:110) e ( − iλH − P ) s Ae ( iλH − P ) s D (cid:111) , (104)where P is given by (93). The parameter λ serves the same purpose as in (51) and Figs. 1 and 2, and the simulationsare performed in a similar fashion as for Fig. 1: The value λ is chosen randomly in a way such that log λ is uniformlydistributed, whereas the Hamiltonians are sampled from the GOE (52). The choice of A and D is more subtle becauseof condition (98). To satisfy this constraint, we rather choose P and A from the Wishart ensemble (53) to subsequentlydetermine D = P + A. The results in Fig. 4 confirm the prediction by the lower bound (103) and show a drastically different behaviourcompared to the fermionic case of Fig. 1. These results can be understood as a manifestation of quantum statistics.However, bosons do not disturb each other statistically when they start piling up (as happens when
Tr( D − A ) ≈ ).Where the fermionic “repulsion” is often more important than the particle interaction, this is not the case for bosons.Hence, the assumption of absence of interactions for bosons is rather more stringent than it is for fermions. Therefore,one should be careful when interpreting these bosonic results when Tr( D − A ) is small and particle densities becomehigh. VII. CONCLUSIONS
We described many-fermion and many-boson systems in which particles are incoherently pumped in and dissipatedfrom the system, such that the total dynamics can be considered to be Markovian (memoryless). We prove that, in9 ������� ������� ● ������ ���������������� �� �� � ������ λ � / � � � � Figure 4. Scatter plot which benchmarks the stationary current J (104) with respect to the minimal current J min (103). Thevariable λ (104) controls the relative strength of the Hamiltonian with respect to incoherent contributions; for each data point, log Λ is randomly chosen from the interval [ − , . The Hamiltonians H in (104) are chosen from the GOE (52) with typicalinteraction v/ √ m between modes, with v = 1 , and mode number m = 10 . The channels P (93) and A (23) in (104) are drawnfrom a Wishart ensemble (53) with m A = 5 and m P = 10 . D in (104) is directly obtained from (93). Data points are comparedto the lower bound J = J min (horizontal red line). The value λ = 1 is indicated (vertical grey line) since it represents thetypical incoherent rate as the mean eigenvalue of P . the absence of interactions between particles, the total particle current across the system exhibits universal propertiesin the stationary state: We could derive an upper bound (50) for fermionic currents, and, under some additionalconditions which prevent the system from unlimited heating, a lower bound (103) for bosonic currents. Remarkably,both bounds are independent of the specific potential landscape the particles are transmitted through.Numerically, we showed that, though counterintuitive, the bounds are typically sharp in the regime where thecoherent dynamics’ frequencies are high compared to the incoherent rates which determine the time scales of thereservoir coupling. This also led us to design Hamiltonians, as generators of the coherent dynamics, which saturatethe bound in the limit where coherent dynamics is dominant. We proved that, in this limit, very general symmetryproperties imposed onto the Hamiltonian suffice to achieve our goal. More specifically, we considered a unitary operatorthat commutes with the Hamiltonian and maps channels which connect the incoming reservoirs to the system ontochannels which connect the system to the outgoing reservoirs. With these design principles , we can saturate ourupper bound for fermionic currents . We note that the centro-symmetry [87–90], discussed in the context of optimaltransport, is a special case of our present design principle. Hence, this work also improves the understanding of howsuch symmetries enhance quantum transport.Our results offer a starting point for the investigation of several new questions, ranging from the relation of the herepresented results to the Landauer formalism [2, 91] to applications, e.g., in the quantum transport theory of disorderedsystems [92–94] or in the quantum statistics of non-equilibrium dynamical processes [95, 96]. On a more fundamentallevel, the natural next steps are to investigate [97] how particle-interactions or other general sources of dephasing [9, 10]impact the here derived universal bounds. In addition, it is a natural question to wonder what happens when theassumption of Markovian dynamics breaks down, e.g., it was recently shown [98], for a non-equilibrium spin-bosonmodel, that the current is optimal for an intermediate coupling between system and reservoirs. ACKNOWLEDGEMENTS
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