No thermalization without correlations
aa r X i v : . [ qu a n t - ph ] O c t No thermalization without correlations
Dmitry V. Zhdanov, ∗ Denys I. Bondar, and Tamar Seideman Northwestern University, Evanston, Illinois 60208, USA Princeton University, Princeton, New Jersey 08544, USA
The proof of the long-standing conjecture is presented that Markovian quantum master equa-tions are at odds with quantum thermodynamics under conventional assumptions of fluctuation-dissipation theorems (implying a translation invariant dissipation). Specifically, except for identifiedsystems, persistent system-bath correlations of at least one kind, spatial or temporal, are obliga-tory for thermalization. A systematic procedure is proposed to construct translation invariant bathmodels producing steady states that well-approximate thermal states. A quantum optical schemefor the laboratory assessment of the developed procedure is outlined.
Introduction . A stochastic interaction of a quantumsystem with a bath brings up the term ˆ F fr in the re-lations for time-dependent expectation values of systemmomenta ˆ p = { ˆ p , . . . , ˆ p N } and positions ˆ x = { ˆ x , . . . , ˆ x N } : ddt h ˆ p n i = − h ∂∂ ˆ x n U (ˆ x ) i + h ˆ F fr n i , (1a) ddt h ˆ x n i = m n h ˆ p n i , (1b)where U (ˆ x ) is a potential energy operator and m k areeffective masses. In this Letter, we study the casewhere ˆ F fr = ˆ F fr (ˆ p ) is position-independent. In this form,Eqs. (1) apply to many quantum phenomena includingthe translational motion of an excited atom in vacuum[1], Brownian motion in a dilute background gas [2],light-driven processes in semiconductor, nanoplasmonicand optomechanical systems [3–5], superconducting cur-rents [6], quantum ratchets [7], energy transport in low-dimensional systems [8], dynamics of chemical reactions[9], two-dimensional vibrational spectroscopy and NMRsignals [10, 11] as well as more exotic entirely quantumdissipative effects [12, 13].The term ˆ F fr (ˆ p ) in Eqs. (1) admits a simple classi-cal interpretation as friction acting on effective parti-cles moving in a potential U ( x ). Such classical dynam-ics are described by the familiar Langevin, Drude andFokker-Plank models when the system-bath interactionsare treated as (i) memoryless (Markovian) and (ii) trans-lation invariant (position-independent). However, wewill show that these two assumptions are at odds withquantum thermodynamics. Specifically, we will provea long-standing no-go conjecture that completely posi-tive Markovian translation-invariant quantum dynamicsobeying Eqs. (1) cannot thermalize.The no-go conjecture was demonstrated by Lindbladas early as in 1976 [15] for a quantum harmonic oscil-lator with a Gaussian damping . Subsequently his par-ticular result was extended to a general quantum sys-tem under the weight of mounting numerical evidence, ∗ [email protected] Positivity of quantum evolution guarantees satisfaction of theHeisenberg uncertainty principle at all times. It was shown thatthe requirements for positivity and complete positivity coincidefor some quantum systems including a harmonic oscillator [14]. The Gaussian damping corresponds to L rel = L lbd µ ˆ x + η ˆ p ( µ , η ∈ however without proof. The no-go conjecture is de-factoincorporated in all popular models such as the Red-field theory [17], the Gaussian phase space ansatz ofYan and Mukamel [18], the master equations of Agar-wal [19], Caldeira-Leggett [20], Hu-Paz-Zhang [21], andLouisell/Lax [22], and the semigroup theory of Lindblad[23] along with specialized extensions in different areasof physics and chemistry. These models break either oneof assumptions (i) and (ii) or the complete positivity ofquantum evolution (see [14, 24, 25] for detailed reviews,note errata [26]). This circumstance is a persistent sourceof controversies (see e.g. the discussions [27–29] of origi-nal works [30, 31]). The matters were further complicatedby the discovery that the free Brownian motion U (ˆ x )=0circumvents the conjecture [32] (we will identify the fullscope of possible exceptions below).The no-go result challenges studies of the long-time dy-namics of open systems. On the one hand, model’s ther-modynamic consistency is undermined by assumptions(i) and (ii). On other hand, the same assumptions openopportunities to simulate large systems that are other-wise beyond the reach. Specifically, the abandonmentof Markovianity entails a substantial overhead to storeand process the evolution history. The value of assump-tion (ii) can be clarified by the following example. Con-sider the re-thermalization of a harmonic oscillator cou-pled to a bath (represented by a collection of harmonicoscillators) after displacement from equilibrium by, e.g.,an added external field, a varied system-bath coupling,or interactions between parts of a compound system.To account for such a displacement without assump-tion (ii), one needs to self-consistently identify the equi-librium position for each bath oscillator, re-thermalizethe bath and modify the system-bath couplings accord-ingly. In practice, this procedure is intractable with-out gross approximations that lead to either numericalinstabilities or physical inaccuracies. Choosing amonga polaron-transformation-based method, Redfield, andF¨orster (hopping) models of quantum transfer epitomizesthis dilemma [33]. C N ) in Eq. (2a) and can be cast to form (3), as shown in Ap-pendix A). The original paper [15] deals with one-dimensionalcase. The multidimensional extension can be found e.g. in [16]. Figure 1. The errors (expressed in the terms of Buresdistance D B between the thermal state ˆ ρ th θ and its approx-imation ˆ ρ st ) in modeling thermal states of a 1D quantumharmonic oscillator in the displaced equilibrium configura-tions (due to a change U (ˆ x ) → U (ˆ x − ∆ x ) in the potential en-ergy) using the conventional quantum optical master equation(dashed lines) and the proposed translation-invariant dissipa-tion model defined by Eqs. (2),(3) and (11) (solid lines). (a)The error dependence on displacement ∆ x for several tem-peratures θ . (b) The error dependence on temperature θ fordifferent values of κ (in units of κ = ~ − β − ). Remarkably, assumption (ii) enables to model the dis-placed state equilibrium by simply adjusting the po-tential energy ˆ U . Fig. 1a shows that without this as-sumption the potential adjustment yields steady stateˆ ρ st significantly different from the canonical equilibriumˆ ρ th θ ∝ e − ˆ Hθ , where θ = k B T and ˆ H is system Hamiltonian.Motivated by these arguments, we propose in this Let-ter a general recipe to construct approximately thermal-izable bath models under assumptions (i) and (ii). Fig. 1illustrates this recipe in application to the above exam-ple. The resulting mismatch between ˆ ρ st and ˆ ρ th θ is small,especially at high temperatures and in the weak system-bath coupling limit. (The calculations details will be ex-plained below.)It will be shown elsewhere that the proposed recipe iscapable of accurately accounting for electronic and spindegrees of freedom. We found it helpful in reservoir en-gineering and optimal control problems. Moreover, theresulting bath models are realizable in the laboratory andcan be used for coupling atoms and molecules nonrecipro-cally [34]. However, the scope of our recipe is limited bythe applicability of assumptions (i) and (ii) and, there-fore, cannot encompass strongly correlated systems (asin the case of Anderson localization [35]). The key results . Starting by formalizing the problem,we write the general master equation that accounts formemoryless system-bath interactions and ensures posi-tivity of the system density matrix ˆ ρ at all times [23]: ∂∂t ˆ ρ = L [ˆ ρ ] , L = L + L rel , (2a) L [ ⊙ ]= i ~ [ ⊙ , ˆ H ] , ˆ H = H (ˆ p , ˆ x )= P Nn =1 ˆ p n m n + U (ˆ x ) , (2b) L rel = K X k =1 L lbdˆ L k , L lbdˆ L [ˆ ρ ] def = ˆ L ˆ ρ ˆ L † − ( ˆ L † ˆ L ˆ ρ +ˆ ρL † ˆ L ) , (2c) where ⊙ is the substitution symbol defined, e.g., in [36].The superoperator L rel accounts for system-bath cou-plings responsible for the friction term ˆ F fr in Eq. (1a)and depends on a set of generally non-Hermitian oper-ators ˆ L k . Based on theorems by A. Holevo [37, 38], B.Vacchini [39–41] has identified the following criterion oftranslational invariance for the L rel : Lemma 1 (The justification is in Appendix A) . Anytranslationally invariant superoperator L rel of the Lind-blad form (2c) can be represented as L rel = P k L lbdˆ A k + L aux with (3a)ˆ A k def = e − i κ k ˆ x ˜ f k (ˆ p ) , L aux = − i [ µ aux ˆ x + f aux (ˆ p ) , ⊙ ] . (3b) where κ k and µ aux are N -dimensional real vectors, ˜ f k arecomplex-valued functions and f aux is real-valued .Theconverse holds as well. The primary findings of this work are summarized inthe following two no-go theorems.
No-go theorem 1.
Let | Ψ i be the ground state (orany other eigenstate of ˆ H ), such that h Ψ | ˆ p | Ψ i =0 ,and which momentum-space wavefunction Ψ ( p )= h p | Ψ i is nonzero almost everywhere, except for some isolatedpoints. Then, no translationally invariant Markovianprocess of form (2) and (3) can steer the system to | Ψ i . The idea of the proof, whose details are given inAppendix B, is to show that the state ˆ ρ = | Ψ ih Ψ | can be the fixed point of superoperator e t L onlyif L rel ≡
0. First, note that the linearity andtranslation invariance of the dissipator (3) implythat L rel [ R g ( x ′ ) e − i ~ x ′ ˆ p ˆ ρ e i ~ x ′ ˆ p d N x ′ ]=0 for any function g ( x ′ ). This equation can be equivalently rewritten as L rel [Ψ (ˆ p ) g (ˆ x )Ψ (ˆ p ) † ]=0 (4)using the identities e − i ~ x ′ ˆ p | Ψ i = √ π ~ Ψ (ˆ p ) | x ′ i and R g ( x ′ ) | x ′ ih x ′ | d N x ′ = g (ˆ x ), where | x ′ i is the eigenstateof position operator: ˆ x k | x ′ i = x ′ k | x ′ i . Let us choose g ( x )= e − i λx , where λ is an arbitrary real vector, andmove to the right the ˆ x -dependent terms in the lhsof Eq. (4) using the commutation relations e − i ˜ λ ˆ x ˆ p =(ˆ p + ~ ˜ λ ) e − i ˜ λ ˆ x with ˜ λ = λ , ± κ k . This rearrangementbrings Eq. (4) to the form ˜ G λ (ˆ p ) e − i λ ˆ x =0 (note that allthe operators of form e ± i ˜ κ k ˆ x expectedly cancel out ow-ing to translation invariance of L rel ). The last equalitycan be satisfied only if the function ˜ G λ ( p ) vanishes iden-tically for all p and λ . However, careful inspection ofAppendix B shows that the latter happens only if L rel =0. The Gaussian dissipators L lbd µ k ˆ x + ˜ f Gk (ˆ p ) ( µ k ∈ R N ) can be reducedto the form Eq. (3) as a limiting case κ k →
0, as shown in Ap-pendix A. The generalized unitary drift term L aux accounts forambiguity of the separation of the quantum Liouvillian L inEq. (2a) into Hamiltonian and relaxation parts. The statement of the 1-st no-go theorem can bestrengthened for a special class of quantum systems. Let B ( p , λ ) be the Blokhintsev function [42], which is relatedto Wigner quasiprobability distribution W ( p , x ) as B ( p , λ )= R ∞−∞ . . . R ∞−∞ e i λx W ( p , x ) d N x . (5) No-go theorem 2.
Suppose that the Blokhintsev func-tion B θ ( p , λ ) of the thermal state ˆ ρ th θ ∝ e − ˆ Hθ characterizedby temperature k B T = θ is such that ∀ p , λ : B θ ( p , λ ) > , B θ ( p , − λ )= B θ ( p , λ ) , (6a) ∀ p = , λ = : B θ ( p , λ )
No translationally invariant Markovianprocess of form (2) and (3) can steer the quantum har-monic oscillator into a thermal state of form ˆ ρ th θ ∝ e − ˆ Hθ .Practical implications of the no-go theorems . In clas-sical thermodynamics, the bath is understood as aconstant-temperature heat tank “unaware” of a systemof interest. However, the no-go theorems indicate thatsystem-bath correlations of at least one kind – spa-tial or temporal – become obligatory for thermalizationonce quantum mechanical effects are taken into account.These correlations break the bath translation invarianceor Markovianity assumptions, respectively.Nevertheless, in the view of computational advantagesoutlined above, it is desirable to incorporate these as-sumptions into the master equations (2) and (3). Now weare going to introduce the recipe to construct such mod-els with a minimal error in the thermal state. In orderto proceed, note that in the limit ( ~ κ k ) ≪ h ˆ p i Eqs. (2)and (3) reduce to the familiar Fokker-Planck equation ∂∂t ̟ ( p ) ≃ Tr[ δ ( p − ˆ p ) L [ˆ ρ ]]+ X n,l ∂ D n,l ( p ) ̟ ( p ) ∂p n ∂p l − X n ∂F fr n ( p ) ̟ ( p ) ∂p n (7)for the momentum probability distribution ̟ ( p )= Tr[ δ ( p − ˆ p )ˆ ρ ]. The friction forces F fr in Eq. (7)as well as Eq. (1a) have the form F fr (ˆ p )= − P k ~ κ k | ˜ f k (ˆ p ) | , (8) whereas the momentum-dependent diffusion operator is D n,l (ˆ p )= ~ P k | ˜ f k (ˆ p ) | κ k,n κ k,l . (9)Equations (8) and (9) can be satisfied by different setsof κ k and ˜ f k ( p ). We will exploit this non-uniquenessto reduce the system-bath correlation errors. Our strat-egy is reminiscent to the familiar way of making den-sity functional calculations practical via error cancella-tion in approximated exchange-correlation functionals.We shall demonstrate the generic procedure by consid-ering a one-dimensional oscillator with the Hamiltonianˆ H = m ˆ p + mω ˆ x (here the dimension subscript n is omit-ted for brevity). Corollary 2.1 implies that L rel [ˆ ρ th θ ] =0and ˆ ρ st =ˆ ρ th θ for any θ , where ˆ ρ st = ˆ ρ | t →∞ is the actualfixed point of the evolution operator e t L . However, thenet discrepancies can be reduced by imposing the follow-ing thermal population conserving constraint: ddt h e − α ˆ H i θ (cid:12)(cid:12)(cid:12) t =0 =0; (cid:12)(cid:12)(cid:12) d dt h e − α ˆ H i θ (cid:12)(cid:12)(cid:12) t =0 → min for all α, (10)where h⊙i θ ( t )= Tr[ ⊙ e t L [ˆ ρ th θ ]]. This constraint can beintuitively justified when the characteristic decay ratesare much smaller than the typical transition frequen-cies, such that the dissipation can be treated per-turbatively. Since the term L rel [ˆ ρ th θ ] generates onlyrapidly oscillating off-diagonal elements in the basisof ˆ H , Eq. (10) ensures that the first-order perturba-tion vanishes on average for the exact thermal state:lim t →∞ t R t e τ L L rel e ( t − τ ) L [ˆ ρ th θ ] d τ =0.In the case of the driftless dissipation L aux =0, Eq. (10)is satisfied by the following functions ˜ f k ( p ) in Eq. (3):˜ f k ( p )= c k e pβ ~ λ k , λ k = κ k tanh( ~ ω θ ) , (11)where β =( m ~ ω ) − and the constants c k should be cho-sen to satisfy Eq. (8). The corresponding dissipator (3)reproduces the familiar microphysical model of quantumBrownian motion (see e.g. Eq. (16) in Ref. [32]) in thelimit κ → ω →
0. Furthermore, the resulting dynamicstends to decrease (increase) the average system energy h ˆ H i θ if its initial temperature θ ′ is higher (lower) than θ : ddt h ˆ H i θ ′ (cid:12)(cid:12) t =0 = c k ω ˜ γ en k ( θ ′ , θ )( h ˆ H i θ − h ˆ H i θ ′ ) (cid:12)(cid:12) t =0 , (12)where ˜ γ en k ( θ ′ , θ )=2 ωβ ~ κ k λ k exp (cid:0) β ~ λ k coth( ~ ω θ ′ ) (cid:1) > ρ st is close to ˆ ρ th θ . Thisconclusion is supported by the simulations presented inFig. 2a for the isotropic dissipator L rel = B κ, ˜ f iso , B κ, ˜ f isodef = L lbdˆ A + + L lbdˆ A − , ˆ A ± = e ∓ iκ ˆ x ˜ f iso ( ± ˆ p ) . (13)One can see that the high-quality thermalization is read-ily achieved by tuning the free parameters c k and κ k evenin the strong dissipation regime.To understand the result (11), note that the terms L lbdˆ A k in Eq. (3) represent independent statistical forces Figure 2. (a) The accuracy of thermalization of the harmonicoscillator at θ =0 by the dissipator L rel =Γ B κ, ˜ f iso as functionof κ and Γ. The solid curves show the Bures distance D B be-tween the thermal state ˆ ρ th θ and its approximation ˆ ρ st for thecase ˜ f iso ( p ) defined by Eq. (11) with c = ω/ p ˜ γ en (0 , f iso ( p ).The dashed curves correspond to the case of functions ˜ f iso ( p )approximated by Eq. (16) with parameters ˜ c i chosen such that d l dp l ( ˜ f iso ( p ) − ˜ g iso ( p )) (cid:12)(cid:12)(cid:12) p =0 =0 for l =0 , ,
2. (b) The Dopplercooling setup to test the model (2), (3) in the laboratory. h− ~ κ k | ˜ f k (ˆ p ) | i contributing to the net friction h ˆ F fr i . Inclassical mechanics, such forces at θ =0 steer the systemto the state of rest by acting against the particles’ mo-menta, hence˜ f k (ˆ p )=0 when pκ k < . (14)However, clipping the functions (11) according toEq. (14) introduces significant errors, as displayed by dot-ted curves in Fig. 2a. Thus, the “endothermic” tails of˜ f k (ˆ p ) at pκ k > ddt h ˆ O i θ =0 for anyobservable ˆ O in the thermodynamic equilibrium ˆ ρ st =ˆ ρ th θ is violated by the master equations (2) and (3) due tothe no-go theorems, i.e., ddt h ˆ x n i θ (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ~ X k h (cid:12)(cid:12) ∂∂ ˆ p n ˜ f k (ˆ p ) (cid:12)(cid:12) i θ (cid:12)(cid:12)(cid:12)(cid:12) t =0 > L aux =0. The inequality (15) pro-vides further evidence for the no-go theorems and is thehallmark of the “position diffusion”, a known artifact inthe quantum theory of Brownian motion [41].According to Eq. (15), ddt h ˆ x i θ (cid:12)(cid:12) t =0 is sensitive tosmoothness of ˜ f k ( p ). Specifically, the rhs of Eq. (15)is exploded by any highly oscillatory components of˜ f k ( p ) and diverges if ˜ f k ( p ) is discontinuous. This en-tirely quantum effect is the origin of poor performanceof the clipped solutions (14) seen in Fig. 2a. Equa-tion (15) uncovers unavoidable errors in the potentialenergy. The optimal solutions (11) enforce error can-cellation ddt h ˆ p m i θ (cid:12)(cid:12) t =0 = − ddt h U (ˆ x ) i θ (cid:12)(cid:12) t =0 between kineticand potential energies leaving the total energy intact ddt h ˆ H i θ (cid:12)(cid:12) t =0 =0. In fact, the error cancellation is achievedwith a large class of physically feasible functions ˜ f k ( p )that may substantially differ from the solutions (11)everywhere but the region of high probability density ̟ ( p )= Tr[ δ (ˆ p − p )ˆ ρ th θ ] (however, note the remark in Ap-pendix D). This is illustrated in Fig. 2a by dashed curvesoverlapping with solid curves.The master equations (2) and (3) provide accuratenon-perturbative description of collisions with a back-ground gas of atoms or photons [5, 40, 44–46]. Hence,the above theoretical conclusions can be directly testedin the laboratory using well-developed techniques, e.g.,the setup shown in Fig. 2b. Here a two-level atomis subject to two orthogonally polarized counterpropa-gating monochromatic nonsaturating laser fields of thesame amplitude E and frequency ω l . We show in Ap-pendix D that the translational motion of the atom canbe modeled using Eq. (2) with an isotropic friction termof form L rel = B κ, ˜ g iso . Here κ = ω l c , ˜ g iso ( p )=˜ c (˜ c +( p − ˜ c ) ) − , ˜ c k ∈ R (16)and the parameters ˜ c k can be tuned by E and ω l .Now we are ready to clarify why the deviations fromcanonical equilibrium increase with | κ | in Fig. 2a. Theparameters ~ | κ | and ˜ g iso ( p ) in Eq. (16) can be regardedas the change of atomic momentum after absorption ofa photon and the absorption rate. The case of small ~ | κ |≪ p h ˆ p i and large ˜ g iso ( p ) implies tiny and frequentmomentum exchanges subject to the central limit the-orem. The net result is a velocity-dependent radiationpressure with vanishing fluctuations. The opposite caseof large ~ | κ |≫ p h ˆ p i and small ˜ g iso ( p ) is the strong shotnoise limit, where the stochastic character of light ab-sorption is no longer averaged out, notably perturbingthe thermal state. Note that a similar interpretation ap-plies to quantum statistical forces in Ref. [47].The dissipative model (2) and (3) with optimized pa-rameters (11) is further analyzed in Fig. 1 using the sameparameters as in Fig. 2a. Both Figs. 1 and 2a indicatethat thermalization can be modeled for a wide rangeof recoil momenta ~ κ ∈ (cid:0) − ( ~ √ β ) − , ( ~ √ β ) − (cid:1) and thehigher the temperature, the better the accuracy. Thus,Eqs. (8) and (9) enable to simulate a variety of velocitydependences of friction and diffusion.Finally, Fig. 1a benchmarks such simulations againstthe commonly used quantum optical master equa-tion (QOME) [48] defined by Eq. (2c) with K =2,ˆ L = √ ω (1 − e − ~ ωθ ) − ˆ a , ˆ L = √ ω ( e ~ ωθ − − ˆ a † , whereˆ a is the harmonic oscillator annihilation operator. Fora correct comparison, the parameters of both modelsare adjusted to ensure identical decay rates in Eq. (12).Systematic errors in our model and QOME are compa-rable for the equilibrium displacements ∆ x ∼ ~ β − atzero temperature and ∆ x ∼ . ~ β − for θ ∼ ~ ω . For low-frequency molecular vibrational modes ( m ∼ atomicunits, ω ∼
200 cm − ), these shifts are of order 0 . .
04 ˚A, respectively, which are in the range of typicalmolecular geometry changes due to optical excitationsor liquid environments. We found the displacement-independent errors in the model (2) and (3) to be veryimportant for quantum control via reservoir engineering.Furthermore, the same feature can also be exploited forengineering the mechanical analogs of nonreciprocal opti-cal couplings [49] and energy-efficient molecular quantumheat machines [34]. These subjects will be explored in aforthcoming publication.
ACKNOWLEDGMENTS
We thank to Alexander Eisfeld for valuable discus-sions and drawing our attention to Refs. [37–41]. Weare grateful to the anonymous referees for important sug-gestions and insightful critic. T. S. and D. V. Zh. thankthe National Science Foundation (Award number CHEM-1012207 to T. S.) for support. D. I. B. is supported byAFOSR Young Investigator Research Program (FA9550-16-1-0254).
SUPPLEMENTAL MATERIAL
INTRODUCTION
This supplemental material is organized as follows. Inthe Sections A, B and C we give the proofs of Lemma 1,first, and second no-go theorems, respectively. Finally,the supporting mathematical derivations for the Dopplercooling model briefly discussed in the main text are pro-vided in Section D.The Roman numbers in parentheses refer everywhereto the equations in the main text of the letter.
Appendix A: The proof of lemma 1
The property of the translational invariance can beformulated as ∀ δx : R δx L rel R − δx = L rel , (A1)where R δx = e − i ~ δx ˆ p ⊙ e i ~ δx ˆ p (A2)is the superoperator of translational shift: ∀ g (ˆ x ) : R ⊺ δx [ g (ˆ x )]= g (ˆ x + δx ).With the help of the canonical commutation relations,any operator ˆ L k = L k (ˆ p , ˆ x ) can expanded in the seriesˆ L k = P l,m c k,l,m ˆ B l,m , where ˆ B l,m = e − i κ l ˆ x g m (ˆ p ) and thefunctions g m ( p ) constitute a set of (not necessarily or-thogonal) basis functions. Using this expansion, any su-peroperator of form L rel = P k L lbdˆ L k can be rewritten as L rel = X k,l ,m ,l ,m c k,l ,m c ∗ k,l ,m ˜ L lbdˆ B l ,m , ˆ B l ,m , (A3)where˜ L lbdˆ A , ˆ A def = ˆ A ⊙ ˆ A † −
12 ( ˆ A ˆ A † ⊙ + ⊙ ˆ A ˆ A † ) . (A4) It follows from Eq. (A1) that if L rel is translationallyinvariant then it should satisfy the identity L rel = 1(2 L ) N Z L − L ... Z L − L R δx L rel R − δx d N d δx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L →∞ = X l X m ,m ˜ c ( l ) m ,m ˜ L lbdˆ B l,m , ˆ B l,m , (A5)where the Hermitian matrices ˜ c ( l ) are defined as˜ c ( l ) m ,m = X k c k,l,m c ∗ k,l,m . (A6)Let us substitute in Eq. (A5) the matrices ˜ c ( l ) with theirJordan decomposition ˜ c ( l ) =˜ u ( l ) ˜ γ ( l ) ˜ u ( l ) † , where ˜ u ( l ) is uni-tary and ˜ γ ( l ) is diagonal. The result is L rel = X l,m L lbdˆ A l,m , (A7)where A l,m = ˜ f l,m (ˆ p ) e − i κ l ˆ x and ˜ f l,m (ˆ p )= q ˜ γ ( l ) m,m × P m ′ ˜ u ( l ) m ′ ,m g m ′ (ˆ p ). Finally, note that Eq. (A7) can becast into the form (3) by replacing the compound index { l, m } with the single consecutive index k . The lemma isproven. Remark 1.
In this work, the Gaussian (continuous)translationally invariant dissipators of form L G = X k L lbdˆ A G k , ˆ A G k = µ k ˆ x + ˜ f Gk (ˆ p ) ( µ k ∈ R N ) (A8)are treated as the limiting case of Eq. (3) with κ k = ǫ µ k →
0. Specifically, one can verify by direct cal-culation that L lbdˆ A G k = L lbdˆ A G k, + + L lbdˆ A G k, − (cid:12)(cid:12)(cid:12)(cid:12) ǫ → − i ~ µ k " ∂ ˜ f Gk (ˆ p ) ∂ ˆ p , ⊙ , (A9)ˆ A G k, ± = √ (cid:16) iǫ ± ˜ f Gk (ˆ p ) (cid:17) e ∓ iǫ µ k ˆ x . (A10) Remark 2.
The translation invariance criterion is gen-eralized to non-Markovian dynamics in Ref. [50].
Appendix B: The proof of no-go theorem 1 (bycontradiction)
Suppose that some eigenstate | Ψ i of Hamiltonian ˆ H is also the fixed point of the quantum Liouvillian L de-fined by Eqs. (2) and (3). Since L rel is assumed transla-tion invariant, it should commute with spatial shift su-peroperator (A2) for any δx . Hence, L rel [ R δx [ˆ ρ ]] = R δx [ L rel [ˆ ρ ]]=0, where ˆ ρ = | Ψ ih Ψ | . Furthermore, thelinearity of L rel implies that ∀ g ( x ′ ) : L rel [ Z g ( x ′ ) R δx [ˆ ρ ] d N x ′ ]=0 . (B1)Equation (B1) can be further simplified using the identity e − i ~ x ′ ˆ p | Ψ i = √ π ~ Ψ (ˆ p ) | x ′ i , (B2)where | x ′ i is the eigenstate of position operator:ˆ x k | x ′ i = x ′ k | x ′ i , h x ′′ | x ′ i = δ ( x ′′ − x ′ ). The validity ofEq. (B2) can be verified by comparing the wave-functions in momentum representation correspondingto its left and right sides. Identities (B2) and R g ( x ′ ) | x ′ ih x ′ | d N x ′ = g (ˆ x ) allow to equivalently rewriteEq. (B1) as ∀ g ( x ′ ) : L rel [ ˆ w g ]=0 , ˆ w g =Ψ (ˆ p ) g (ˆ x )Ψ (ˆ p ) † . (B3)Consider the case g ( x )= g λ ( x )= e − i λx , where λ is somereal N -dimensional vector. Note that the operator L rel [ ˆ w g ] then includes the explicit dependence on coor-dinate operators ˆ x only in forms of matrix exponentials e − i λ ˆ x , e ± i κ k ˆ x and commutators [ˆ x k , ⊙ ]. Using the com-mutation relation e − i ˜ λ ˆ x ˆ p =(ˆ p + ~ ˜ λ ) e − i ˜ λ ˆ x with ˜ λ = λ , ± κ k ,it is possible to group out the momentum and coordinateoperators in L rel [ ˆ w g ] and rewrite the condition (B3) as:0= L rel [ ˆ w g λ ]= ˜ G λ (ˆ p ) e − i λ ˆ x , (B4)where ˜ G λ ( p )= G ( p , p + ~ λ )Ψ ( p )Ψ ( p + ~ λ ) ∗ (B5)and G ( p , p ′ )= X k (cid:16) F k ( p ) F k ( p ′ ) ∗ − | ˜ f k ( p ) | + | ˜ f k ( p ′ ) | (cid:17) + (B6) ~ µ aux ( ∂ ln(Ψ ( p )) ∂ p + ∂ ln(Ψ ( p ′ ) ∗ ) ∂ p ′ ) − i ( f aux ( p ) − f aux ( p ′ )) ,F k ( p )= ˜ f k ( p + ~ κ k ) Ψ ( p + ~ κ k )Ψ ( p ) . (B7)Note that the cancellation of all the matrix exponentials e ± i κ k ˆ x at the rhs of Eq (B4) is the consequence of thetranslation invariance of L rel . Condition (B4) impliesthat ∀ p ∈ R N , ∀ λ ∈ R : ˜ G λ ( p )=0 , (B8) and ∀ p , p ′ ∈ R N : G ( p , p ′ )=0 (except a possible zero mea-sure subset of points { p , p ′ } where Ψ (ˆ p )Ψ (ˆ p ′ ) † =0). Inparticular, this means that ∀ n, ∀ p , p ′ ∈ R N : ∂ ∂p n ∂p ′ n G ( p , p ′ )= X k ∂F ( n ) k ( p ) ∂p n (cid:18) ∂F ( n ) k ( p ′ ) ∂p ′ n (cid:19) ∗ =0 . (B9)Equality (B9) can be satisfied only if ∀ k : F k ( p ) ∝ const, i.e., if ˜ f k ( p )= c k Ψ ( p − ~ κ k )Ψ ( p ) , where c k is some real constant. Substitution ofthis expression and λ = into Eq. (B5) gives˜ G ( p )= P k c k (cid:0) | Ψ ( p ) | −| Ψ ( p − ~ κ k ) | (cid:1) + ~ µ aux ∂ | Ψ ( p ) | ∂ p . Multiplication of the both sides of Eq. (B8) by p and subsequent integration over p gives: Z p ˜ G ( p ) d N p = X k c k ~ κ k − ~ ( µ aux + X k c k κ k ) h Ψ | ˆ p | Ψ i =0 . (B10)According to our assumption, h Ψ | ˆ p | Ψ i = . Hence,Eq. (B10) implies that P k c k | κ k | =0. This equalityholds only if ∀ k : κ k = . However, in this case allfunctions ˜ f k ( p )= c k reduce to constants, so that L rel =0.This result completes the proof. Appendix C: The proof of no-go theorem 2 (bycontradiction)
Denote as Ψ k ( p ) and E k ( k =0 , ..., ∞ ) the momentum-space wavefunction and energy of the k -th eigenstate | Ψ k i of the Hamiltonian ˆ H . The thermal state ˆ ρ th θ can beexpressed in these notations asˆ ρ th θ = ˜ N X k e − Ekθ | Ψ k ih Ψ k | (C1)Suppose that there exists such relaxation superopera-tor of form (3) that L rel [ˆ ρ th θ ]=0. Owing to assumed lin-earity and translational invariance of L rel , the thermalstate ˆ ρ th θ should satisfy the relation similar to (B1): ∀ g ( x ′ ) : L rel [ Z g ( x ′ ) R δx [ˆ ρ th θ ] d N x ′ ]=0 , (C2)where R δx is the spatial shift superoperator defined byEq. (A2). With the help of relation (B2), one can apply Using (A9), it is straightforward to deduce that the corre-sponding summand for Gaussian dissipator (A8) takes form ~ µ ∂∂ p ( µ ∂∂ p | Ψ ( p ) | ). The resulting contribution in the lhsof Eq. (B10) is | µ | ~ . The equality h Ψ | ˆ p | Ψ i = holds for any non-degenerate eigen-state of the time-reversal invariant Hamiltonian (2b) to Eq. (C2) the same procedure as was used to derive theequality (B3) from Eq. (B1). The result is ∀ g ( x ) : L rel [ ˆ w θ,g ]=0 , (C3)where ˆ w θ,g = ˜ N X k e − Ekθ Ψ k (ˆ p ) g (ˆ x )Ψ k (ˆ p ) † . (C4)Consider the case g ( x )= g λ ( x )= e − i λx , where λ is somereal N -dimensional vector. The result of application of L rel to ˆ w θ,g λ can be represented after some algebra as L rel [ ˆ w θ,g λ ]= G (cid:0) ˆ p + ~ λ , λ (cid:1) e − i λ ˆ x , (C5)where G ( p , λ ) = − iB θ ( p , λ ) (cid:0) f aux ( p − ~ λ ) − f aux ( p + ~ λ ) (cid:1) + ~ µ aux ∂B θ ( p , λ ) ∂ p + X k (cid:18) Q k,n ( p + ~ κ k , λ ) − B θ ( p , λ ) (cid:18)(cid:12)(cid:12)(cid:12) ˜ f k (cid:0) p + ~ λ (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ f k (cid:0) p − ~ λ (cid:1)(cid:12)(cid:12)(cid:12) (cid:19)(cid:19) , (C6) Q k ( p , λ ) = B θ ( p , λ ) ˜ f k ( p − ~ λ ) ˜ f ∗ k ( p + ~ λ ) . (C7)In derivation of (C6) the identity B ( p , λ )= ˜ N X k e − Ekθ Ψ k ( p − ~ λ )Ψ ∗ k ( p + ~ λ ) (C8)was used which follows directly from the definition (5) ofthe Blokhintsev function.Eqs. (C3) and (C5) require that ∀ p , λ : G ( p , λ )=0 , (C9)and hence ∀ λ : ¯ G ( λ )= R ∞−∞ . . . R ∞−∞ d N p G ( p , λ )=0,where G ( p , λ )= G ( p , λ )+ G ( p , − λ )= X k (cid:26) − (cid:12)(cid:12)(cid:12) ˜ f k (cid:0) p + ~ λ (cid:1) − ˜ f k (cid:0) p − ~ λ (cid:1)(cid:12)(cid:12)(cid:12) B θ ( p , λ )+ X α,β = ± βQ k (cid:16) p + β +12 ~ κ k , α λ (cid:17)(cid:27) +2 ~ µ aux ∂B θ ( p , λ ) ∂ p . (C10)The last equality in (C10) is obtained assuming that B θ ( p , − λ )= B θ ( p , λ ) (see Eq. (6a)). It is easy to checkthat the integrations over all terms in the last line of(C10) cancel out, so that¯ G ( λ )= − Z ∞−∞ . . . Z ∞−∞ d N p × X k (cid:12)(cid:12)(cid:12) ˜ f k (cid:0) p + ~ λ (cid:1) − ˜ f k (cid:0) p − ~ λ (cid:1)(cid:12)(cid:12)(cid:12) B ( p , λ ) . (C11) According to the assumption (6a), the integrand in(C11) is nonnegative. Moreover, ¯ G ( λ )=0 iif ∀ k :˜ f k ( p )= c k =const. Hence, the expression (C6) for G ( p , λ ) can be simplified as G ( p , λ ) = X k c k ( B ( p + ~ κ k , λ ) − B ( p , λ )) . (C12)Note that the terms L lbdˆ A k in Eq. (3) with ˜ f k ( p )=const willhave non-trivial effect only if κ k =0 . However, it followsfrom (6b) that in this case G ( , ) < Appendix D: Testing the model (2) and (3) in thelaboratory
In this section, we provide the detailed analysis of theDoppler cooling example introduced in the main text (seeFig. 2b in the main text) and prove that the coolingmechanism is the quantum friction of form (13).In the proposed setup an atom is subject to two orthog-onally polarized counterpropagating beams of the samefield amplitude E and carrier frequency ω l (hereafter inthis section we will omit the subscript l for shortness sinceit will not cause any ambiguity). We assume that ω isclose to the frequency ω a of the transition g ↔ e betweenthe ground g and degenerate excited e electron states of s - and p -symmetries, respectively. Let d be the abso-lute value of the transition dipole moment and γ be theexcited state spontaneous decay rate.For the spatial arrangement depicted in Fig. 2b thetranslation motion of the atom along x -axis is coupled tothe field-induced electron dynamics since each absorbedor coherently emitted photon changes the x -componentof atomic momentum hereafter denoted as p . Further-more, we will assume that the spontaneous decay doesnot affect the x -component of atomic momentum. Thelatter condition can be achieved using, e.g., an arrange-ment shown in Fig. 3.The master equation which describes this coupled dy-namics can be written within the rotating wave approx-imation in the form (2) withˆ H = ˆ p m − ~ ω a | g ih g | + (cid:26) ξ ( t ) | e ih g | e − i ( ωt − κ ˆ x ) + ξ ( t ) | e ih g | e − i ( ωt + κ ˆ x ) +h.c. (cid:27) (D1) In the case of Gaussian dissipator (A8) Eq. (C12) reduces to G ( p , λ ) = 12 ~ X k,m,n µ k,n µ k,m ∂ ∂p n p m B ( p , λ ) . (C12*)By assumption (6b), the quadratic form ∂ ∂p n p m B ( p , λ ) in(C12*) is negative-definite at { p , λ } = { , } . Hence, G ( , ) < Figure 3. The possible Doppler cooling setup where stochas-tic recoil accompanying the spontaneous emission is dampedalong the x -axis. Here the atom of interest A is put into inter-sected orthogonal optical cavities formed by pairs of mirrors M , M ′ and M , M ′ . The cavities are tuned resonant to theatomic g ↔ e transition and force atom to spontaneously emitabsorbed photons predominantly in the directions perpendic-ular to the x -axis via the Purcell effect. The decay rate γ canbe controlled by changing the cavities Q-factors. The collat-eral increase of the energy of motions along y - and z -axes isrestricted by sympathetic cooling by two auxiliary atoms B and C . and L rel = γ X n =1 L lbd | g ih e n | . (D2)Here ξ k ( t )= − ~d k ~ E k ( t ), where ~d and ~d are the transi-tion dipole moments associated with the s → p z and s → p y electronic transitions into degenerate electronically ex-cited sublevels e and e , respectively, and ~ E k ( t ) is theslowly varying complex amplitude of the associated fieldcomponent. The remaining notations are defined in themain text.The mean value of any observable of form ˆ O = f (ˆ p, ˆ x )can be written in Heisenberg representation as: h ˆ O ( t ) i = Tr[ˆ ρ U L t,t ⊺ [ ˆ O ]] , (D3)where we define: ∀ L ( t ) : U L t,t def = ⇒ T e R tt = t L d t . (D4)The symbol ⇒ T in (D4) denotes the chronological order-ing superoperator which arranges operators in direct (in-verse) time order for t>t ( t 1. Weak coherent laser driving In this regime, ξ ( t )= ξ ( t )= ξ =const, and there existssuch δt in the range of applicability of the second-orderexpansion (D14) that δt ≫ γ − . Thence, the integrals in(D19) can be easily computed, which gives: L effrel = B κ, ˜ f iso , ˜ f iso ( p )= | ξ | ~ p γ/ q ( γ ) +∆ ( − p ) , (D26)ˆ H eff = − | ξ | ~ X α = ± ∆ ( α ˆ p )( γ ) + ∆ ( α ˆ p ) . (D27)Note what the Hamiltonian ˆ H eff describes the effect ofthe optical quadratic Stark shift which also can inducethe effective potential forces on the system in the case ofspatially non-uniform fields ξ = ξ ( x ). 2. Incoherent driving Suppose that the the atom is illuminated by the twoclassical light sources with the equal spectral densities I ( ω ) at the atomic site and having coherence timesin the range ∆ − ( p ) ≪ t coh ≪ γ − . In this case, ξ ( t ) and ξ ( t ) represent the uncorrelated stationary stochas-tic processes. This allows one to choose such δt , that γ − ≫ δt ≫ t coh , and calculate the integrals in Eqs. 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