The quantum-to-classical transition and decoherence

Transcription

1 The quantum-to-classical transition an ecoherence Maximilian Schlosshauer Department of Physics, University of Portlan, 5 North Willamette Boulevar, Portlan, Oregon 9723, USA I give a peagogical overview of ecoherence an its role in proviing a ynamical account of the quantum-to-classical transition. The formalism an concepts of ecoherence theory are reviewe, followe by a survey of master equations an ecoherence moels. I also iscuss methos for mitigating ecoherence in quantum information processing an escribe selecte experimental investigations of ecoherence processes. To appear in: M. Aspelmeyer, T. Calarco, J. Eisert, an F. Schmit-Kaler (es.), Hanbook of Quantum Information (Springer: Berlin/Heielberg, 214). Realistic quantum systems are never completely isolate from their environment. When a quantum system interacts with its environment, it will in general become entangle with a large number of environmental egrees of freeom. This entanglement influences what we can locally observe upon measuring the system. In particular, quantum interference effects with respect to certain physical quantities (most notably, classical quantities such as position) become effectively suppresse, making them prohibitively ifficult to observe in most cases of practical interest. This is the process of ecoherence, sometimes also calle ynamical ecoherence or environment-inuce ecoherence [1 9]. State in general an interpretation-neutral terms, ecoherence escribes how entangling interactions with the environment influence the statistics of results of future measurements on the system. Formally, ecoherence can be viewe as a ynamical filter on the space of quantum states, singling out those states that, for a given system, can be stably prepare an maintaine, while effectively excluing most other states, in particular, nonclassical superposition states of the kin popularize by Schröinger s cat. In this way, ecoherence lies at the heart of the quantum-to-classical transition. It ensures consistency between quantum an classical preictions for systems observe to behave classically. It provies a quantitative, ynamical account of the bounary between quantum an classical physics. In any concrete experimental situation, ecoherence theory specifies the physical requirements, both qualitative an quantitative, for pushing the quantum classical bounary towar the quantum realm. Decoherence is a pure quantum effect, to be istinguishe from classical issipation an stochastic fluctuations (noise). Decoherence processes are extremely efficient. Even when the environment oes not, from a classical point of view, impart significant classical perturbations on the system, quantum-mechanically the system will in most circumstances become rapily an strongly entangle with the environment. Furthermore, ue to the many uncontrollable egrees of freeom of the environment, such entanglement is usually irreversible for all practical purposes. Increasingly realistic moels of ecoherence proarxiv: v1 [quant-ph] 9 Apr 214 CONTENTS I. Introuction 1 II. Basic formalism an concepts 2 A. Decoherence an interference amping 2 B. Environmental monitoring an information transfer 3 C. Environment-inuce superselection an ecoherence-free subspaces 4 1. Pointer states an the commutativity criterion 5 2. Decoherence-free subspaces 6 D. Proliferation of information an quantum Darwinism 6 E. Decoherence versus issipation an noise 7 III. Master equations 7 A. Born Markov master equations 8 B. Linbla master equations 8 C. Non-Markovian ecoherence 9 IV. Decoherence moels 1 A. Collisional ecoherence 1 B. Quantum Brownian motion 11 C. Spin boson moels 13 D. Spin-environment moels 14 V. Qubit ecoherence, quantum error correction, an error avoiance 14 A. Correction of ecoherence-inuce quantum errors 14 B. Quantum computation on ecoherence-free subspaces 15 C. Environment engineering an ynamical ecoupling 16 VI. Experimental stuies of ecoherence 16 A. Atoms in a cavity 17 B. Matter-wave interferometry 17 C. Superconucting systems 18 VII. Decoherence an the founations of quantum mechanics 19 References 19 I. INTRODUCTION

2 2 cesses have been evelope, progressing from toy moels to complex moels tailore to specific experiments (see Sec. IV). Avances in experimental techniques have mae it possible to observe the graual action of ecoherence in experiments such as matter-wave interferometry [1], cavity QED [11], an superconucting systems [12] (see Sec. VI). The superposition states necessary for quantum information processing are typically also those most susceptible to ecoherence. Thus, ecoherence is a major barrier to implementing evices for quantum information processing such as quantum computers (see Sec. V). Qubit systems must be engineere to minimize environmental interactions etrimental to the preparation an longevity of the esire superposition states. At the same time, they must remain sufficiently open to allow for their control. Quantum error correction can uno some of the ecoherence-inuce egraation of the superposition state an will be an integral part of quantum computers (see Sec. V A). Not only is ecoherence relevant to quantum information, but also vice versa. An information-centric view of quantum mechanics proves helpful in conveying the essence of the ecoherence process an is also use in recent explorations of the role of the environment as an information channel (see Sec. II B). It is a curious historical accient (Joos s term [13, p. 13]) that the role of the environment in quantum mechanics was appreciate only relatively late. While one can fin for example, in Heisenberg s writings [14] a few early anticipatory remarks about the role of environmental interactions in the quantum-mechanical escription of systems, it wasn t until the 197s that the ubiquity an implications of environmental entanglement were realize by Zeh [1, 15]. It took another ecae for the formalism of ecoherence to be evelope, chiefly by Zurek [2, 3], an for concrete moels an numerical estimates of ecoherence rates to be worke out [16, 17]. Review papers on ecoherence inclue Refs. [4 6, 18]. There are two books on ecoherence: a volume by Joos et al. [8] (a collection of chapters written by ifferent authors) an a monograph by this author [9]. Ref. [19] also contains material on ecoherence. Founational implications of ecoherence are iscusse in Refs. [6, 7, 9, 2]. II. BASIC FORMALISM AND CONCEPTS In the ouble-slit experiment, we cannot observe an interference pattern if we also measure which slit the particle went through (that is, if we obtain perfect which-path information). In fact, there is a continuous traeoff between interference (phase information) an which-path information: the better we can istinguish the two possible paths, the less visible the interference pattern becomes [21]. What is more, for a ecrease in interference visibility to occur it suffices that there are egrees of freeom somewhere in the worl that, if they were measure, woul allow us to make, with a certain egree of confience, a statement about the path of the particle through the slits. While we cannot say that prior to their measurement, these egrees of freeom have encoe information about a particular, efinitive path of the particle instea, we have merely correlations involving both possible paths no actual measurement is require to bring about the ecrease in interference visibility. It is enough that, in principle, we coul make such a measurement to obtain which-path information. This is somewhat loose talk, an conceptual caveats lurk. But it captures quite well the essence of what is happening in ecoherence, where those egrees of freeom somewhere in the worl are the egrees of freeom of the system s environment interacting with the system, leaing to the creation of quantum correlations (entanglement) between system an environment. Decoherence can thus be thought of as a process arising from the continuous monitoring of the system by the environment; effectively, the environment is performing nonemolition measurements on the system (see Sec. II B). We now give a formal quantum-mechanical account of what we have just trie to convey in wors, an then flesh out the consequences an etails. A. Decoherence an interference amping Consier again the ouble-slit experiment an enote the quantum states of the particle (call it S, for system ) corresponing to passage through slit 1 an 2 by s 1 an s 2, respectively. Suppose that the particle interacts with another system E for example, a etector or an environment such that if the quantum state of the particle before the interaction is s 1, then the quantum state of E will become E 1 (an similarly for s 2 ), resulting in the final composite states s 1 E 1 an s 2 E 2, respectively. For an initial superposition state α s 1 +β s 2, the final composite state will be entangle, Ψ = α s 1 E 1 + β s 2 E 2. (1) The statistics of all possible local measurements on S are exhaustively encoe in the reuce ensity matrix ρ S, ρ S = Tr E (ρ SE ) = Tr E Ψ Ψ = α 2 s 1 s 1 + β 2 s 2 s 2 + αβ s 1 s 2 E 2 E 1 + α β s 2 s 1 E 1 E 2. (2) For example, suppose we measure particle s position by letting the particle impinge on a istant etection screen. Statistically, the resulting particle probability ensity p(x) will be given by p(x) = Tr S (ρ S x) = α 2 ψ 1 (x) 2 + β 2 ψ 2 (x) Re {αβ ψ 1 (x)ψ 2(x) E 2 E 1 }, (3)

3 3 where ψ i (x) x s i. The last term represents the interference contribution. Thus, the visibility of the interference pattern is quantifie by the overlap E 2 E 1, i.e., by the istinguishability of E 1 an E 2. In the limiting case of perfect istinguishability, E 2 E 1 =, no interference pattern will be observable an we obtain the classical preiction. Phase relations have become locally (i.e., with respect to S) inaccessible, an there is no measurement on S that can reveal coherence between s 1 an s 2. The coherence is now between the states s 1 E 1 an s 2 E 2, requiring an appropriate global measurement (acting jointly on S an E) for it to be reveale. Conversely, if the interaction between S an E is such that E is completely unable to resolve the path of the particle, then E 1 an E 2 are inistinguishable an full coherence is retaine at the level of S, as is also irectly obvious from Eq. (1). In the intermeiary regime where < E 2 E 1 < 1, meaning that E 1 an E 2 can be istinguishe in a one-shot measurement with nonzero probability p = 1 E 2 E 1 2 < 1, an interference pattern of reuce visibility is obtaine. Equation (3) shows that the reuction in visibility increases as E 1 an E 2 become more istinguishable. Here is another way of putting the matter. Looking back at Eq. (1), we see that E encoes which-way information about S in the same relative-state sense [22] in which EPR correlations [23 25] may be sai to encoe information. That is, if E 2 E 1 = an we were to measure E an foun it to be in state E 1, we coul, in EPR s wors [23, p. 777], preict with certainty that we will fin S in s 1. 1 Whenever such a preiction is possible were we to measure E, no interference effects between the components s 1 an s 2 can be measure at S, even if E is never actually measure. If E 2 E 1 >, then E encoes only partial which-way information about S, in the sense that a measurement of E coul not reliably istinguish between E 1 an E 2 ; instea, sometimes the measurement will result in an outcome compatible with both E 1 an E 2. Consequently, an interference experiment carrie out on S woul fin reuce visibility, representing iminishe local coherence between the components s 1 an s 2. As hinte above, the escription evelope so far escribes the essence of the ecoherence process if we ientify the particle S more generally with an arbitrary quantum system an the secon system E with the environment of S. Then an iealize account of the ecoherence 1 Of course, this must not be rea as saying that S was alreay in s 1 (i.e., went through slit 1 ) prior to the measurement of E. Nor oes it mean that the result of a subsequent path measurement on S is necessarily etermine, by virtue of the measurement on E, prior to this S-measurement s actually being carrie out. After all, as Peres has cautione us [26], unperforme measurements have no outcomes. So while the picture of E as encoing which-path information about S is certainly suggestive an helpful, it shoul be use with an unerstaning of its conceptual pitfalls. interaction has form ( ) c i s i E i i c i s i E i (t). (4) We have here introuce a time parameter t, where t = correspons to the onset of the environmental interaction, with E i (t) E for all i; at t < the system an environment are assume to be uncorrelate (an assumption common to most ecoherence moels). A single environmental particle interacting with the system will typically only insufficiently resolve the components s i in the system s superposition state. But because of the large number of such particles (an, hence, egrees of freeom), the overlap between their ifferent joint states E i (t) will rapily ecrease as a result of the buil-up of many interaction events. Specifically, in many ecoherence moels an exponential ecay of overlap is foun [3, 5, 9, 16, 19, 27 3], E i (t) E j (t) e t/τ for i j. (5) Here τ is the characteristic ecoherence timescale, which can be evaluate for particular choices of the parameters in each moel (see Sec. IV). B. Environmental monitoring an information transfer We will now motivate, in a ifferent an more rigorous way, the picture of ecoherence as a process of environmental monitoring. First, we express the influence of the environment in a completely general way. We assume that at t = there are no correlations between system S an environment E, ρ SE () = ρ S () ρ E (). We write ρ E () in its iagonal ecomposition, ρ E () = i p i E i E i, where i p i = 1 an the states E i form an orthonormal basis of the Hilbert space of E. 2 If H enotes the Hamiltonian (here assume to be timeinepenent) of SE an U(t) = e iht represents the unitary time evolution operator, then the ensity matrix of S evolves accoring to [ ( )] } ρ S (t) = Tr E {U(t) ρ S () p i E i E i U (t) = ij i p i E j U(t) E i ρ S () E i U (t) E j. (6) Introucing the Kraus operators [31] efine by W ij pi E j U(t) E i, we obtain ρ S (t) = ij E ij ρ S ()E ij. (7) 2 In cases where the environment oes not start out in a pure state, we can always purify it through the introuction of an aitional (fictitious) environment. Without loss of generality, we can therefore always take the environment to be in a pure state before its interaction with the system.

4 4 It is customary to combine the two inices i an j into a single inex an write the Kraus operators as such that W k p ik E jk U(t) E ik, (8) ρ S (t) = k W k ρ S ()W k. (9) This Kraus-operator formalism (also calle operator-sum formalism) represents the effect of the environment as a sequence of (in general nonunitary) transformations of ρ S generate by the operators W k. The Kraus operators exhaustively encoe information about the initial state of the environment an about the ynamics of the joint SE system. Because the evolution of SE is unitary, the Kraus operators satisfy the completeness constraint W k W k = I S, (1) k where I S is the ientity operator in the Hilbert space of S. Equations (9) an (1) together imply that the W k are the generators of a completely positive map Φ : ρ S () ρ S, also known as a quantum operation [31] or quantum channel. 3 We will now use Eq. (9) to formally motivate the view that ecoherence correspons to an inirect measurement of the system by the environment, an that it thus results from a transfer of information from the system to the environment (see also Ref. [18]). In such an inirect measurement, we let the system S interact with a probe here the environment E followe by a projective measurement on E. The probe is treate as a quantum system. This proceure aims to yiel information about S without performing a projective (an thus estructive) irect measurement on S. To moel such an inirect measurement, consier again an initial composite ensity operator ρ SE () = ρ S () ρ E () evolving uner the action of U(t) = e iht, where H is the total Hamiltonian. Consier a projective measurement M on E with eigenvalues α an corresponing projectors P α α α, with Pα 2 = P α = P α. The probability of obtaining outcome α in this measurement when S is escribe by the ensity operator ρ S (t) is Prob (α ρ S (t)) = Tr E (P α ρ E (t)) = Tr E { Pα Tr S [ U(t) (ρs () ρ E ()) U (t) ]}. (11) 3 The Kraus formalism is of limite use in calculating ecoherence ynamics for concrete situations of physical interest. This is so because fining the Kraus operators correspons to iagonalizing the full Hamiltonian of SE, usually a prohibitively ifficult task. Moreover, the Kraus operators contain all contributions to the evolution of the reuce ensity matrix, while for consierations of ecoherence we are typically intereste only in the nonunitary terms, an certain contributions such as backaction effects from the system on the environment can often be neglecte. (This is where master equations come into play; see Sec. III.) The ensity matrix of S conitione on the particular outcome α is ρ (α) S (t) = Tr E {[I P α ] ρ SE (t) [I P α ]} Prob (α ρ S (t)) = Tr { E [I Pα ] U(t) [ρ S () ρ E ()] U (t) [I P α ] }. Prob (α ρ S (t)) (12) Inserting the iagonal ecomposition ρ E () = k p k E k E k an carrying out the trace gives [18] ρ (α) S (t) = k M α,k ρ S (t)m α,k Prob (α ρ S (t)). (13) Here we have introuce the measurement operators M α,k p k α U(t) E k, (14) which obey the completeness constraint α,k M α,km α,k = I S. Equation (12) escribes the effect of the inirect measurement on the state of the system. If, however, we o not actually inquire about the result of this measurement, we must assign to the system a ensity operator that is a sum over all the possible conitional states ρ (α) S (t) weighte by their probabilities Prob (α ρ S (t)), ρ S (t) = α Prob (α ρ S (t)) ρ (α) S (t) = α,k M α,k ρ S (t)m α,k. (15) Note that this expression is formally analogous to the Kraus-operator expression of Eq. (9), which escribe the effect of a general environmental interaction on the state of the system. Recall, further, that the situation we encounter in ecoherence is precisely one in which we o not actually rea out the environment or, in the present picture, in which we o not inquire about the result of the inirect measurement. This suggests that ecoherence can inee be unerstoo as an inirect measurement a monitoring of the system by its environment. C. Environment-inuce superselection an ecoherence-free subspaces Decoherence can occur in any basis; which observable is monitore by the environment epens on the specific form of the system environment interaction. The preferre states (or preferre observables) of the system emerge ynamically as those states that are the most robust to the interaction with the environment, in the sense that they become least entangle with the environment; thus, they are the states most immune to ecoherence.

5 5 This is the stability criterion for the selection of preferre states, resulting in the ynamical selection of preferre states ( environment-inuce superselection ) [1 3, 15]. These environment-superselecte preferre states (or observables) are sometimes also calle pointer states (or pointer observables) [2], since they correspon to the physical quantities that are most easily rea off at the level of the system, akin to the pointer on the ial of a measurement apparatus. 1. Pointer states an the commutativity criterion To fin the preferre states, we ecompose the total system environment Hamiltonian into the self- Hamiltonians of the system S an environment E representing the intrinsic ynamics, an a part H int representing the interaction between system an environment, H = H S + H E + H int. (16) In many cases of practical interest, H int ominates the evolution of the system, H H int (the quantummeasurement limit of ecoherence). We look for system states s i such that the composite system environment state, when starting from a prouct state s i E at t =, remains in the prouct form s i E i (t) for all t > uner the action of H int (we shall assume here that H int is not explicitly time-epenent). That is, we eman that (setting 1 from here on) e ihintt s i E = λ i s i e ihintt E s i E i (t). (17) Thus, the pointer states s i are the eigenstates of the part of the interaction Hamiltonian H int pertaining to the Hilbert space of the system, with eigenvalues λ i. These states will be stationary uner H int [2]. It follows that the pointer observable efine by O S = i o i s i s i commutes with H int, [ ] OS, H int =. (18) This commutativity criterion [2, 3] is particularly easy to apply when H int takes the tensor-prouct form H int = S E, as is frequently the case. Then the environmentsuperselecte observables will be those observables that commute with S. If S is Hermitian, it represents the physical quantity monitore by the environment. In general, any H int can be written as a iagonal ecomposition of (unitary but not necessarily Hermitian) system an environment operators S α an E α, H int = α S α E α. If the S α are Hermitian, such a Hamiltonian represents the simultaneous environmental monitoring of ifferent observables S α of the system. A sufficient conition for { s i } to form a set of pointer states of the system is then given by the requirement that the s i be simultaneous eigenstates of the operators S α, S α s i = λ (α) i s i for all α an i. (19) Interaction Hamiltonians frequently escribe the scattering of surrouning particles (photons, air molecules, etc.), leaing to collisional ecoherence (see Sec. IV A). Since the force laws escribing such processes typically epen on some power of istance, the interaction Hamiltonian will then commute with the position operator. Thus, the pointer states will be approximate eigenstates of position (i.e., narrow position-space wave packets). This explains why superpositions of mesoscopically an macroscopically istinct positions are prohibitively ifficult to observe [2, 3, 16, 3, 32 38]. Collisional ecoherence can also be ominant in microscopic systems when these systems occur in istinct spatial configurations that couple strongly to the surrouning meium. For example, chiral molecules such as sugar are always observe to be in chirality eigenstates (left-hane or right-hane), which are superpositions of ifferent energy eigenstates. Any attempt to prepare such molecules in energy eigenstates leas to immeiate ecoherence into the environmentally stable chirality eigenstates [39, 4]. The quantum limit of ecoherence [41] arises when the moes of the environment are slow in comparison with the evolution of the system that is, when the highest frequencies (i.e., energies) available in the environment are smaller than the separation between the energy eigenstates of the system. Then the environment will be able to monitor only quantities that are constants of motion. In the case of nonegeneracy, this quantity will be the energy of the system, leaing to the environment-inuce superselection of energy eigenstates for the system [41]. 4 In many realistic situations, the commutativity criterion, Eq. (18), can only be fulfille approximately [42, 43]. In aition, the self-hamiltonian of the system an the interaction Hamiltonian may contribute in roughly equal strengths (e.g., in moels for quantum Brownian motion [4, 44]; see Sec. IV B), renering neither the quantummeasurement limit of negligible intrinsic ynamics nor the quantum limit of ecoherence of a slow environment appropriate. In such cases, more general methos for etermining the preferre states are require. The preictability-sieve strategy [42, 43, 45] computes the time epenence of the amount of ecoherence introuce into the system for a large set of initial states of the system evolving uner the total system environment Hamiltonian. Typically, this ecoherence is measure using either the purity Tr ( ρs) 2 or the von Neumann entropy 4 Textbooks on quantum mechanics usually attribute a special role to such energy eigenstates (for close systems) since they are stationary uner the action of the Hamiltonian. In this closesystem picture, however, arbitrary superpositions of energy eigenstates shoul nonetheless be perfectly legitimate. Thus, it is important to realize that the environment-inuce superselection of energy eigenstates is not equivalent to a situation in which the presence of the environment coul be neglecte altogether; instea, the environment plays the crucial role of continuously monitoring the energy of the system, leaing to a local suppression of coherence between energy eigenstates.

6 6 S(ρ S ) = Tr (ρ S log 2 ρ S ) of the reuce ensity matrix ρ S. The states most immune to ecoherence will be those which lea to the smallest ecrease in purity or the smallest increase in von Neumann entropy. Application of this metho leas to a ranking of the possible preferre states with respect to their robustness to the interaction with the environment. For particular moels it has been explicitly shown that the states picke out by the preictability sieve are robust to the particular choice of the measure of ecoherence. For example, in the moel for quantum Brownian motion, ifferent measures lea to the same minimum-uncertainty wave packets in phase space [5, 8, 15, 43, 46, 47]. 2. Decoherence-free subspaces The pointer-state conition of Eq. (19) can be strengthene to the concept of pointer subspaces [3] or ecoherence-free subspaces (DFS) [48 57]. These are subspaces of the Hilbert space of the system in which every state in the subspace is immune to ecoherence; this is a nontrivial requirement, since in general superpositions of pointer states will not be pointer states themselves. One important conition for this to happen is that the preferre states s i efine by Eq. (19) form an orthonormal basis of the subspace, an that the eigenvalues λ (α) i in Eq. (19) are inepenent of i, i.e., that all s i are simultaneous egenerate eigenstates of each S α, S α s i = λ (α) s i for all α an i. (2) This conition states that the action of a given S α must be the same for all basis states s i of the DFS, an thus the existence of a DFS correspons to a symmetry in the structure of the system environment interaction, i.e., to a ynamical symmetry. A necessary conition for such a symmetry to obtain is the absence of terms in H int that act jointly on system an environment in a nontrivial manner. An arbitrary state ψ in the DFS can then be written as ψ = i c i s i an will evolve accoring to e ihintt ψ E = ψ e i( α λ(α) E α)t E ψ E ψ (t). (21) Thus, the state ψ oes not become entangle with the environment an is therefore immune to ecoherence. When the self-hamiltonian H S of the system cannot be neglecte, one nees to aitionally ensure that none of the basis states s i of the DFS will rift out of the subspace uner the evolution generate by H S. Otherwise an initially ecoherence-free state woul again become prone to ecoherence. The concept of DFS can be generalize to the formalism of noiseless subsystems (or noiseless quantum coes) [57 59]. D. Proliferation of information an quantum Darwinism Quantum Darwinism [6 68] buils on the ieas of ecoherence an environmental encoing of information, by broaening the role of the environment to that of a communication an amplification channel. Interactions between the system an its environment lea to the reunant storage of selecte information about the system in many fragments of the environment. By measuring some of these fragments, observers can inirectly obtain information about the system without appreciably isturbing the system itself. Inee, this represents how we typically observe objects. For example, we see an object not by irectly interacting with it, but by intercepting scattere photons that encoe information about the object s spatial structure [66, 67]. In this sense, quantum Darwinism provies a ynamical explanation for the robustness of states of (especially) macroscopic objects to observation. It was foun that the observable of the system that can be imprinte most completely an reunantly in many istinct fragments of the environment coincies with the pointer observable selecte by the system environment interaction [61 64]; conversely, most other states o not seem to be reunantly storable. Inee, it has been shown that the reunant proliferation of information regaring pointer states is as inevitable as ecoherence itself [69]. Quantum Darwinism has been stuie in several concrete moels, for example, in spin environments [63], quantum Brownian motion [7], an photon an photon-like environments [66, 67, 69]. The efficiency of the amplification process escribe by quantum Darwinism can be expresse in terms of the quantum Chernoff information [69]. The structure an amount of information that the environment encoes about the system can be quantifie using the measure of (classical [61, 62] or quantum [5, 63, 64]) mutual information. Classical mutual information is base on the choice of particular observables of the system S an the environment E an quantifies how well one can preict the outcome of a measurement of a given observable of S by measuring some observable on a fraction of E [61, 62]. Quantum mutual information is efine as S(ρ S )+S(ρ E ) S(ρ), where ρ S, ρ E, an ρ are the ensity matrices of S, E, an the composite system SE, respectively, an S(ρ) = Tr (ρ log 2 ρ) is the von Neumann entropy associate with ρ. Quantum mutual information quantifies the egree of quantum correlations between S an E. Classical an quantum mutual information give similar results [5, 61 64] because the ifference between the two measures, known as the quantum iscor [71], isappears when ecoherence is sufficiently effective to select a well-efine pointer basis [71].

7 7 E. Decoherence versus issipation an noise While the presence of issipation implies the presence of ecoherence, the converse is not necessarily true. When issipation an ecoherence are both present, they typically occur on vastly ifferent timescales; the ecoherence timescale is typically many orers of magnitue shorter than the relaxation timescale. A rule-of-thumb estimate for the ratio of the relaxation timescale τ r to the ecoherence timescale τ for a massive object escribe by a superposition of two ifferent positions a istance x apart is [17] τ r τ ( ) 2 x, (22) λ B where λ B = (2mkT ) 1/2 is the thermal e Broglie wavelength of the object. For an object of mass m = 1 g at room temperature in a coherent superposition of two locations a istance x = 1 cm apart, τ r /τ is on the orer of 1 4. Thus, for macroscopic objects the issipative influence of the environment is usually completely negligible on the timescale relevant to the ecoherence inuce by this environment. Decoherence is a consequence of environmental entanglement. In the literature on quantum computing, however, the term ecoherence is often use to refer to any process that affects the qubits, incluing perturbations ue to classical fluctuations an imperfections. Examples for sources of such classical noise in the context of quantum computing are the fluctuations in the intensity [72] an uration [73] of the laser beam incient on qubits in an ion trap, inhomogeneities in the magnetic fiels in NMR quantum computing [74], an bias fluctuations in superconucting qubits [75]. The istinction between classical noise an quantum ecoherence has been further blurre in quantum error correction, since the error-correcting schemes are insensitive to the physical origin of the qubit errors (see Sec. V A). Phenomenologically an formally the influence of classical noise processes may be escribe in a manner similar to the effect of environmental entanglement, namely, in terms of a ecay of the off-iagonal elements (interference terms) in the local ensity matrix (in the environment-superselecte basis). But in the case of noise, the ecay of the off-iagonal elements occurs because the system s ensity matrix is ientifie with an average over a physical ensemble of systems (or, put ifferently, over the ifferent instances of particular noise processes), while in the case of ecoherence the ecay is ue to an entanglement-inuce elocalization of phase coherence for iniviual systems. The funamental ifference between these physical processes is maske by the ensity-matrix escription. Inee, one can always fin an experimental proceure that woul, at least in principle, istinguish between the ifferent physical processes unerlying formally similar ensity-matrix escriptions. In contrast with ecoherence, noise oes not create system environment entanglement an can in principle always be unone using only local operations (witness, for example, the reversal of ensemble ephasing in NMR experiments using the spin-echo technique). In any iniviual realization of the noise process the ynamics of the system are completely unitary, an thus no coherence is lost from the system. By contrast, if the system becomes entangle with environmental egrees of freeom, at the very least we woul nee to perform a pair of measurements on the environment before an after the interaction with the system in orer to gather enough information to reverse the effect of ecoherence by application of an appropriate countertransformation. Moreover, as also seen experimentally [76], these measurements woul not always constitute a sufficient proceure for unoing ecoherence (see also Sec. IV.C of Ref. [5]). The loss of phase coherence ue to environmental entanglement is sometimes simulate (with the above caveats) by classical fluctuations perturbing the system, i.e., by the aition of certain time-epenent terms to the self-hamiltonian of the system. This strategy was implemente, for example, in theoretical [72, 77] an experimental [76, 78] stuies of the influence of fluctuating parameters in ion-trap quantum computers. III. MASTER EQUATIONS In the usual approach to moeling ecoherence, the reuce ensity matrix ρ S (t) is obtaine from ρ S (t) = Tr E ρ SE (t) Tr E { U(t)ρSE ()U (t) }, (23) where U(t) is the time-evolution operator for the composite system SE. The task of calculating ρ SE (t) is often computationally cumbersome or even intractable. It is also unnecessarily etaile, because we are usually only intereste in the ynamics of the system. A master equation allows us to calculate ρ S (t) irectly from an expression of the form ρ S (t) = V(t)ρ S (), (24) where the superoperator V(t) is the ynamical map generating the evolution of ρ S (t). If the master equation is exact, { then we merely have the ientity V(t)ρ S () Tr E U(t)ρSE ()U (t) } an no computational avantage is gaine. Therefore, master equations are typically base on simplifying approximations. In moeling ecoherence, we focus on master equations that are first-orer time-local ifferential equations of the form t ρ S(t) = L [ρ S (t)] i [H S, ρ S (t)] + D[ρ S (t)]. (25) This equation is local in time in the sense that the change of ρ S at time t epens only on ρ S evaluate at t. The superoperator L acting on ρ S (t) typically epens on the initial state of the environment an the ifferent terms in the Hamiltonian. We have ecompose L into two

8 8 parts to istinguish their physical interpretation. The first term, i [H S, ρ S(t)], is unitary an given by the Liouville von Neumann commutator with the renormalize Hamiltonian H S of the system. (Because the environment typically leas to a renormalization of the energy levels of the system, this Hamiltonian oes in general not coincie with the unperturbe free Hamiltonian H S of S that woul generate the evolution of S in absence of the environment.) The secon, nonunitary term D[ρ S (t)] represents ecoherence (an often also issipation) ue to the environment. A. Born Markov master equations Born Markov master equations allow for many ecoherence problems to be treate in a mathematically simple, an often close, form. They are base on the following two approximations: 1. The Born approximation. The system environment coupling is sufficiently weak an the environment is reasonably large such that changes of the ensity operator ρ E of the environment are negligible an the system environment ensity operator remains remains approximately factorize at all times, ρ SE (t) ρ S (t) ρ E. 2. The Markov approximation. Memory effects of the environment are negligible, in the sense that any self-correlations within the environment create by the coupling to the system ecay rapily compare with the characteristic timescale over which the state of the system varies noticeably. Comparisons between the preictions of moels base on Born Markov master equations an experimental ata inicate that the Born an Markov assumptions are reasonable in many physical situations (but see Sec. III C below for exceptions an non-markovian moels). Assuming these assumptions hol an writing the interaction Hamiltonian as H int = α S α E α, the Born Markov master equation reas [9, 19] t ρ S(t) = i [H S, ρ S (t)] α {[S α, B α ρ S (t)] + [ρ S (t)c α, S α ]}, (26) where the system operators B α an C α are efine as B α C α τ β τ β c αβ (τ)s (I) β ( τ), (27a) c βα ( τ)s (I) β ( τ). (27b) Here S α (I) ( τ) enotes the operator S α in the interaction picture. In the following, we will simplify notation by omitting the superscript I ; instea we use the convention that all operators bearing explicit time arguments are to be unerstoo as interaction-picture operators. (For ensity operators, however, we will maintain the superscript notation in orer to istinguish them from Schröinger-picture ensity operators, which also carry a time argument.) The quantities c αβ (τ) appearing in Eq. (27) are given by c αβ (τ) E α (τ)e β ρe. (28) These environment self-correlation functions quantify how much information the environment retains over time about its interaction with the system. The Markov approximation correspons to the assumption of a rapi ecay of the c αβ (τ) relative to the timescale set by the evolution of the system. In many situations of interest, the general form of the Born Markov master equation, Eq. (26), simplifies consierably. For example, typically only a single system observable S is monitore by the environment, H int = S E. Also, the time epenence of the operators S α (τ) an E α (τ) is often simple, facilitating the calculation of the quantities B α an C α. Examples are iscusse in Sec. IV. B. Linbla master equations Linbla master equations constitute a particular, albeit quite general, class of time-local Markovian master equations. They arise from the requirement that the master equation must ensure the positivity of the reuce ensity matrix at all times, i.e., that ψ ρ S (t) ψ for all t an any pure state ψ of the system S. This requirement is physically reasonable, since we woul like to interpret the iagonal elements ψ ρ S (t) ψ as occupation probabilities. While the positivity conition is automatically fulfille if the evolution is exact, approximate master equations will not necessarily ensure positivity of ρ S (t) [79]. The Linbla master equation is a special case of the general Born Markov master equation that ensures positivity an takes the general form [8, 81] t ρ S(t) = i [H S, ρ S (t)] + 1 ] [ ]} γ αβ {[S α, ρ S (t)s β + S α ρ S (t), S β, (29) 2 αβ where the coefficients γ αβ are time-inepenent an exhaustively encapsulate information about the physical parameters of the ecoherence processes (an possibly issipation processes). 5 Note that a Born Markov master equation that cannot be written in Linbla form 5 It is possible to bring any Born Markov master equation into Linbla form by imposing the rotating-wave approximation.

9 9 oes not necessarily violate positivity. For example, the exact, non-markovian master equation for quantum Brownian motion cannot be brought into Linbla form without imposing several approximations an assumptions [82]; yet, because it is exact, it ensures positivity [27, 83]. The Linbla master equation, Eq. (29), can be simplifie by iagonalizing the coefficient matrix Γ (γ αβ ). This iagonalization is always possible because Γ is positive, i.e., all its eigenvalues κ µ are. Then Eq. (29) takes the iagonal form [81, 84] t ρ S(t) = i [H S, ρ S (t)] 1 { κ µ L 2 µ L µ ρ S (t) + ρ S L µl µ 2L µ ρ S (t)l µ}. µ (3) Here H S is the renormalize Hamiltonian of the system. The Linbla operators L µ are linear combinations of the original operators S α, with coefficients etermine by the iagonalization of Γ. Because the S α are not necessarily Hermitian, the Linbla operators o not always correspon to physical observables. But when they o, we can rewrite Eq. (3) in compact ouble-commutator form, t ρ S(t) = i [H S, ρ S (t)] 1 2 κ µ [L µ, [L µ, ρ S (t)]]. (31) As an example, consier a situation in which the environment monitors the position of a system. With L = x an the free -particle Hamiltonian H S = H S = p 2 /2m, Eq. (31) becomes t ρ S(t) = i [ p 2, ρ S (t) ] 1 2m 2 κ [x, [x, ρ S(t)]]. (32) Expressing this master equation in the position representation results in ρ S (x, x, t) t µ = i ( ) 2 2 2m x 2 x 2 ρ S (x, x, t) 1 2 κ (x x ) 2 ρ S (x, x, t). (33) This is the classic equation of motion for ecoherence ue to environmental scattering first erive in Ref. [16]. Linbla master equations provie an intuitive an simple way of representing the environmental monitoring of an open quantum system. Most of the real physics behin this monitoring process is hien in the coefficients This assumption, ubiquituous in quantum optics, is justifie whenever the typical timescale for the evolution of the system is short in comparison with the relaxation timescale of the system. (See Sec of Ref. [19] for etails.) κ µ appearing in Eq. (3). If the Linbla operators are chosen to be imensionless, the κ µ can be irectly interprete as ecoherence rates, since they have units of inverse time. Equation (31) shows that the ecoherence term vanishes if [L µ, ρ S (t)] = for all µ, t. (34) In this case, ρ S (t) evolves unitarily. Since the L µ are linear combinations of the S α, Eq. (34) typically means that [S α, ρ S (t)] = for all α, t. This implies that simultaneous eigenstates of all S α will be immune to ecoherence, which is precisely the pointer-state criterion of Eq. (19). In quantum-jump an quantum-trajectory approaches, the evolution of the reuce ensity matrix is conitione on an explicitly observe sequence of measurement results in the environment. This allows for the (formal) escription of a single realization of the system evolving stochastically, conitione on a particular measurement recor. The ynamics are then escribe by a master equation of the Linbla type, Eq. (31), for the reuce ensity matrix ρ C S conitione on the measurement recors of the Linbla operators L µ, ρ C S = i [ H S, ρ C ] 1 S t 2 + µ [ κ µ Lµ, [ L µ, ρ C ]] S t µ κµ W[L µ ]ρ C S W µ. (35) Here, W[L]ρ Lρ+ρL ρ Tr { Lρ + ρl }, an the W µ enote so-calle Wiener increments. Equation (35) correspons to a iffusive unraveling of the Linbla equation into iniviual quantum trajectories, which can then be expresse by means of a stochastic Schröinger equation [85 97]. C. Non-Markovian ecoherence The erivation of the Born Markov master equation assumes that the coupling between system an environment is weak an memory effects of the environment can be neglecte. These conitions, however, are not met in certain situations of physical interest. An example woul be a superconucting qubit strongly couple to a low-temperature environment of other two-level systems [98, 99]. Also, a recent experiment [1] has measure strongly non-ohmic spectral ensities for the environment of a quantum nanomechanical system; such ensities lea to non-markovian evolution. In many cases, pronounce memory effects in the environment will cause strong epenencies of the evolution of the reuce ensity operator on the past history of the system environment composite an therefore make it impossible to escribe the reuce ynamics by a ifferential equation that is local in time. Surprisingly, however,

10 1 one can show that even non-markovian ynamics sometimes can still be escribe by a time-local ifferential equation of the form t ρ S(t) = K(t)ρ S (t), (36) where the superoperator K(t) epens only on t. For example, a non-markovian master equation for quantum Brownian motion (see Sec. IV B) can be obtaine through a formal moification of the Born Markov master equation [4, 5]. In general, it is often possible to arrive at non-markovian but time-local master equations via the so-calle time-convolutionless projection operator technique [11 14]. IV. DECOHERENCE MODELS Many physical systems can be represente either by a qubit if the state space of the system is iscrete an effectively two-imensional, or by a particle escribe by continuous phase-space coorinates. Neeless to say, in the case of quantum information processing the qubit representation is of particular relevance. Similarly, a wie range of environments can be moele as a collection of quantum harmonic oscillators or qubits. Harmonic-oscillator environments are of great generality. At low energies, many systems interacting with an environment can effectively be represente by one or two coorinates of the system linearly couple to an environment of harmonic oscillators; inee, sufficiently weak interactions with an arbitrary environment can be mappe onto a system linearly couple to a harmonic-oscillator environment [15, 16]. Environments represente by qubits are often the appropriate moel in the low-temperature regine, where ecoherence is typically ominate by interactions with localize moes, such as paramagnetic spins, paramagnetic electronic impurities, tunneling charges, efects, an nuclear spins [98, 99, 17]. Each of the localize moes is represente by a finite-imensional Hilbert space with a finite energy cutoff. We can therefore moel these moes as a set of iscrete states. Typically, only two such states are relevant, an thus the localize moes can be mappe onto an environment of qubits. Since qubits can be formally represente by spin- 1 2 particles, such moels are known as spin-environment moels. In the following, we will iscuss four important stanar moels, namely, collisional ecoherence (Sec. IV A), quantum Brownian motion (Sec. IV B), the spin boson moel (Sec. IV C), an the spin spin moel (Sec. IV D). For etails on these an other ecoherence moels, incluing erivations of the relevant master equations, see Secs. 3 an 5 of Ref. [9]. A. Collisional ecoherence Collisional ecoherence arises from the scattering of environmental particles by a massive free quantum particle. Moels of collisional ecoherence were first stuie in the classic paper by Joos an Zeh [16]. A more rigorous an general treatment was later evelope by Hornberger an collaborators [3, 35 38] (see also [33, 34, 18]), which, among other refinements, remeie a flaw in Joos an Zeh s original erivation that ha resulte in ecoherence rates that were too large by a factor of 2π [3]. If we assume that the central particle is much more massive than the environmental particles such that its center-of-mass state is not isturbe by the scattering events (no recoil), the time evolution of the reuce ensity matrix is given by [9, 16, 3, 33, 34] ρ S (x, x, t) t = F (x x )ρ S (x, x, t). (37) This master equation escribes pure spatial ecoherence without issipation. The ecoherence factor F (x x ) plays the role of a localization rate. It represents the characteristic ecoherence rate at which spatial coherences between two positions x an x become locally suppresse an is given by F (x x ) = q ϱ(q)v(q) ˆn ˆn ( 1 e iq(ˆn ˆn ) (x x ) ) f(qˆn, qˆn ) 2. (38) 4π Here ϱ(q) enotes the number ensity of incoming particles with magnitue of momentum equal to q = q, ˆn an ˆn are unit vectors (with ˆn an ˆn representing the associate soli-angle ifferentials), an v(q) enotes the spee of particles with momentum q. For the scattering of massive environmental particles we have v(q) = q/m, where m is each particle s mass, while for the scattering of photons an other massless particles v(q) is equal to the spee of light. The quantity f(qˆn, qˆn ) 2 is the ifferential cross section for the scattering of an environmental particle from initial momentum q = qˆn to final momentum q = qˆn. Whenever the mass of the central particle becomes comparable to the mass of the environmental particles (as in the case of air molecules scattere by small molecules an free electrons [19]), the no-recoil assumption oes not hol an more general moels for collisional ecoherence have to be consiere [34, 35]. The resulting ynamics inclue issipation, as well as ecoherence in both position an momentum. To further evaluate the ecoherence factor F (x x ), Eq. (38), we istinguish two important limiting cases. In the short-wavelength limit, the typical wavelength of the scattere environmental particles is much shorter than the coherent separation x = x x between the welllocalize wave packets in the spatial superposition state

11 11 of the system. Then a single scattering event will be able to fully resolve this separation an thus carry away complete which-path information, leaing to maximum spatial ecoherence per scattering event. In this limit, F (x x ) turns out to be simply equal to the total scattering rate Γ tot [9]. This implies the existence of an upper limit for the ecoherence rate when increasing the separation x, in contrast with ecoherence rates obtaine from linear moels [compare Eqs. (22) an (54)]. Equation (37) then shows that spatial interference terms will become exponentially suppresse at a rate set by Γ tot, ρ S (x, x, t) = ρ S (x, x, )e Γtott. (39) In the opposite long-wavelength limit, the environmental wavelengths are much larger than the coherent separation x = x x, which implies that an iniviual scattering event will reveal only incomplete which-path information. For this case, one can show that spatial coherences become exponentially suppresse at a rate that epens on the square of the separation x [9], ρ S (x, x, t) = ρ S (x, x, )e Λ( x)2t, (4) where Λ is a scattering constant that encapsulates the physical etails of the interaction. Thus, the quantity Λ( x) 2 plays the role of a ecoherence rate. The epenence of this rate on x is reasonable: if the environmental wavelengths are much larger than x, it will require a large number of scattering events to encoe an appreciable amount of which-path information in the environment, an this amount will increase, for a given number of scattering events, as x becomes larger. Note that if x is increase beyon the typical wavelength of the environment, the short-wavelength limit nees to be consiere instea, for which the ecoherence rate is inepenent of x an attains its maximum possible value. Numerical values of collisional ecoherence rates obtaine from Eqs. (39) an (4), with the physically relevant scattering parameters Γ tot an Λ appropriately evaluate, have shown the extreme efficiency of collisions in suppressing spatial interferences; Table I shows a few classic orer-of-magnitue estimates [8, 9, 16]. Excellent agreement between theory an experiment has been emonstrate for the ecoherence of fullerenes ue to collisions with backgroun gas molecules in a Talbot Lau interferometer [3, ] (see Sec. VI B an Fig. 2), an for the ecoherence of soium atoms in a Mach Zehner interferometer ue to the scattering of photons [114] an gas molecules [115]. TABLE I. Estimates of ecoherence timescales (in secons) for the suppression of spatial interferences over a istance x equal to the size a of the object ( x = a = 1 3 cm for a ust grain an x = a = 1 6 cm for a large molecule). See Ref. [9] for etails. Environment Dust grain Large molecule Cosmic backgroun raiation Photons at room temperature Best laboratory vacuum Air at normal pressure is given by H E = i ( 1 p 2 i + 1 ) 2m i 2 m iωi 2 qi 2, (41) where m i an ω i enote the mass an natural frequency of the ith oscillator, an q i an p i are the canonical position an momentum operators. The interaction Hamiltonian H int escribes the bilinear coupling of the system s position coorinate x to the positions q i of the environmental oscillators, H int = x i c iq i, where the c i enote coupling strengths. This interaction represents the continuous environmental monitoring of the position coorinate of the system. The Born Markov master equation escribing the evolution of the ensity matrix ρ S (t) of the system is given by [9, 44] t ρ S(t) = i [ H S, ρ S (t) ] τ ( ν(τ) [ x, [ x( τ), ρ S (t) ]] iη(τ) [ x, { x( τ), ρ S (t) }]). (42) Here, x(τ) enotes the system s position operator in the interaction picture, x(τ) = e ihsτ xe ihsτ. The curly brackets {, } in the secon line enote the anticommutator {A, B} AB + BA. The functions ( ω ) ν(τ) = ω J(ω) coth cos (ωτ), (43) 2kT η(τ) = ω J(ω) sin (ωτ), (44) are known as the noise kernel an issipation kernel, respectively. The function J(ω), calle the spectral ensity of the environment, is given by B. Quantum Brownian motion J(ω) i c 2 i 2m i ω i δ(ω ω i ). (45) A classic an extensively stuie moel of ecoherence an issipation is the one-imensional motion of a particle weakly couple to a thermal bath of noninteracting harmonic oscillators, a moel known as quantum Brownian motion. The self-hamiltonian H E of the environment In general, spectral ensities encapsulate the physical properties of the environment. One frequently replaces the collection of iniviual environmental oscillators by an (often phenomenologically motivate) continuous function J(ω) of the environmental frequencies ω.

12 12 If we specialize to the important case of the system represente by a harmonic oscillator with self-hamiltonian p p H S = 1 2M p MΩ2 x 2, (46) the resulting Born Markov master equation is t ρ S(t) = i [ H S M Ω 2 x 2, ρ S (t) ] iγ [ x, { p, ρ S (t) }] D [ x, [ x, ρ S (t) ]] f [ x, [ p, ρ S (t) ]]. (47) The coefficients Ω 2, γ, D, an f are efine as x x Ω 2 2 M γ 1 MΩ D f 1 MΩ τ η(τ) cos (Ωτ), τ η(τ) sin (Ωτ), τ ν(τ) cos (Ωτ), τ ν(τ) sin (Ωτ). (48a) (48b) (48c) (48) The first term on the right-han sie of Eq. (47) represents the unitary ynamics of a harmonic oscillator whose natural frequency is shifte by Ω. The secon term escribes momentum amping (issipation) at a rate proportional to γ, which epens only on the spectral ensity but not the temperature of the environment. The thir term is of the Linbla ouble-commutator form [see Eq. (31)] an escribes ecoherence of spatial coherences over a istance X at a rate D( X) 2. Note that D epens on both the spectral ensity J(ω) an the temperature T of the environment. The fourth term also represents ecoherence, but its influence on the ynamics of the system is usually negligible, especially at higher temperatures. In the long-time limit γt 1, the master equation (47) escribes ispersion in position space given by X 2 (t) = D 2m 2 t. (49) γ2 That is, the with X(t) of the ensemble in position space asymptotically scales as X(t) t, just as in classical Brownian motion; hence the term quantum Brownian motion. Figure 1 shows the time evolution of position-space an momentum-space superpositions of two Gaussian wave packets in the Wigner picture, as escribe by Eq. (47) [27]. Interference between the two wave packets is represente by oscillations between the irect peaks. The interaction with the environment amps these oscillations. The amping occurs on ifferent timescales for the two initial conitions. While the momentum coorinate is not irectly monitore by the environment, the intrinsic ynamics, through their creation of spatial superpositions from superpositions of momentum, result FIG. 1. Evolution of superpositions of Gaussian wave packets in quantum Brownian motion as stuie in Ref. [27], visualize in the Wigner representation. Time increases from top to bottom. In the left column, the initial wave packets are separate in position; in the right column, the separation is in momentum. in ecoherence in momentum space. This interplay of environmental monitoring an intrinsic ynamics leas to the emergence of pointer states that are minimumuncertainty Gaussians (coherent states) well-localize in both position an momentum, thus approximating classical points in phase space [5, 8, 15, 27, 43, 46, 47]. Let us consier the important case of an ohmic spectral ensity J(ω) ω with a high-frequency cutoff Λ, J(ω) = 2Mγ π ω Λ 2 Λ 2 + ω 2. (5) In the limit of a high-temperature environment (kt Ω an kt Λ), we arrive at the Caleira Leggett master equation [116], t ρ S(t) = i [ H S, ρ S (t) ] [ { iγ x, p, ρs (t) }] where 2Mγ kt [ x, [ x, ρ S (t) ]], (51) H S = H S M Ω 2 x 2 = 1 2M p M [ Ω 2 2γ Λ ] x 2 (52)

13 13 is the frequency-shifte Hamiltonian H S of the system. This equation has been wiely an successfully use to moel ecoherence an issipation processes, even in cases where the assumptions were not strictly fulfille (for example, in quantum-optical settings, where often kt Λ [117]). In the position representation, the final term on the right-han sie of Eq. (51) can be written as ( ) x x 2 γ ρ S (x, x, t), (53) λ B where λ B = (2MkT ) 1/2 is the thermal e Broglie wavelength. This term escribes spatial localization with a ecoherence rate τ 1 x x given by [17] τ 1 x x = γ ( ) x x 2. (54) λ B This is Eq. (22), an as iscusse there, given that λ B is extremely small for macroscopic an even mesoscopic objects, we see that superpositions of macroscopically separate center-of-mass positions will typically be ecohere on timescales many orers of magnitue shorter than the issipation (relaxation) timescale γ 1. Over timescales on the orer of the ecoherence time, we may therefore often neglect the issipative term in Eq. (51), leaing to the pure-ecoherence master equation t ρ S(t) = i [ H S, ρ S (t) ] 2Mγ kt [ x, [ x, ρ S (t) ]]. (55) C. Spin boson moels In the spin boson moel, a qubit interacts with an environment of harmonic oscillators. The seminal review paper by Leggett et al. [28] iscusses the ynamics of the spin boson moel in great etail. Let us first consier a simplifie spin boson moel where the self-hamiltonian of the system is taken to be H S = 1 2 ω σ z, with eigenstates an 1. In contrast with the more general case iscusse below, this Hamiltonian oes not inclue a tunneling term 1 2 σ x, an thus H S oes not generate any nontrivial intrinsic ynamics. We employ the familiar self-hamiltonian, Eq. (41), for an environment of harmonic oscillators, an choose the bilinear interaction Hamiltonian H int = σ z i c iq i. Using the raising an lowering operators a an a, we can recast the total Hamiltonian as H = 1 2 ω σ z + ω i a i a i + σ z ) (g i a i + g i a i. (56) i i Note that since [ ] H, σ z =, no transitions between an 1 can be inuce by H. There is no energy exchange between the system an the environment, an we therefore eal with a moel of ecoherence without issipation. Such a moel is a goo representation of rapi ecoherence processes uring which the amount of issipation is negligible, as is often the case in physical applications. The resulting evolution can be solve exactly [9]. For an ohmic spectral ensity with a high-frequency cutoff, it is foun that superpositions of the form α +β 1 are exponentially ecohere on a timescale set by the thermal correlation time (kt ) 1 of the environment. Inclusion of a tunneling term 1 2 σ x yiels the general spin boson moel efine by the Hamiltonian H = 1 2 ω σ z 1 2 σ x + i ( 1 p 2 i + 1 ) 2m i 2 m iωi 2 qi 2 + σ z i c i q i. (57) The rich non-markovian ynamics of this moel have been analyze in Refs. [28, 118]. The particular ynamics strongly epen on the various parameters, such as the temperature of the environment, the form of the spectral ensity (subohmic, ohmic, or supraohmic), an the system environment coupling strength. For each parameter regime, a characteristic ynamical behavior emerges: localization, exponential or incoherent relaxation, exponential ecay, an strongly or weakly ampe coherent oscillations [28]. In the weak-coupling limit, one can erive the Born Markov master equation in much the same way as in the case of quantum Brownian motion (note the similar structure of the Hamiltonians). The result is (see Ref. [9] for etails) ( ) t ρ S(t) = i H Sρ S (t) ρ S (t)h S D [σ z, [σ z, ρ S (t)]] + ζσ z ρ S (t)σ y + ζ σ y ρ S (t)σ z. (58) The first term on the right-han sie of the master equation (58) represents the evolution uner the environmentshifte self-hamiltonian H S, the secon term correspons to ecoherence in the σ z eigenbasis of the system at a rate given by D, an the last two terms escribe the ecay of the two-level system. H S is the renormalize (an in general non-hermitian) Hamiltonian of the system. The coefficients ζ, D, f, an γ are given by ζ = f i γ, D = f = γ = τ ν(τ) cos ( τ), τ ν(τ) sin ( τ), τ η(τ) sin ( τ), (59a) (59b) (59c) (59) with the noise an the issipation kernels ν(τ) an η(τ) taking the same form as in quantum Brownian motion [see Eqs. (43) an (44)].

14 14 D. Spin-environment moels A qubit linearly couple to a collection of other qubits known also as a spin spin moel is often a goo moel of a single two-level system, such as a superconucting qubit, strongly couple to a low-temperature environment [98, 99]. The moel of a harmonic oscillator interacting with a spin environment may be relevant to the escription of ecoherence an issipation in quantumnanomechanical systems an micron-scale ion traps [119]. For etails on the theory of spin-environment moels, see Refs. [99, ]. A simple version of a spin spin moel is escribe by the total Hamiltonian H = H S + H int = 1 2 σ x σ z N i=1 g i σ (i) z 1 2 σ x σ z E. (6) Here, H S represents the intrinsic ynamics given by a tunneling term, while H int escribes the environmental monitoring of the observable σ z. The moel can be solve exactly [123, 124], an the resulting ynamics illustrate the epenence of the preferre basis on the relative strengths of the self- Hamiltonian of the system an the interaction Hamiltonian. The preferre basis emerges as the local basis that is most robust uner the total Hamiltonian. When the interaction Hamiltonian ominates over the self-hamiltonian, the pointer states are foun to be eigenstates of the interaction Hamiltonian, in agreement with the commutativity criterion, Eq. (18). Conversely, when the moes of the environment are slow an the self- Hamiltonian ominates the evolution of the system (the quantum limit of ecoherence [41]), the pointer states are the eigenstates of the Hamiltonian of the system. In the weak-coupling limit, spin environments can be mappe onto oscillator environments [15, 125]. Specifically, the reuce ynamics of a system weakly couple to a spin environment can be escribe by the system couple to an equivalent oscillator environment escribe by an explicitly temperature-epenent spectral ensity of the form ( ω ) J eff (ω, T ) J(ω) tanh, (61) 2kT where J(ω) is the original spectral ensity of the spin environment. (See Sec of Ref. [9] for etails an examples.) V. QUBIT DECOHERENCE, QUANTUM ERROR CORRECTION, AND ERROR AVOIDANCE Quantum computation an quantum information processing rely on coherent superpositions of mesoscopically or macroscopically istinct states that are highly susceptible to ecoherence. Avoiing, controlling, an mitigating ecoherence is therefore of paramount importance. While the qubits nee to be protecte from etrimental environmental interactions, we also nee to be able to control an measure them via a macroscopic apparatus. The formiable challenge of esigning a quantum computer consists of meeting both emans in a balance way. Even so, ecoherence inuce by interactions with the environment an the control apparatus, as well as noise ue to faulty gate operations, will likely be too strong to allow for useful quantum computations to be carrie out [73, 126]. What is also neee is an active mitigation of the effects of ecoherence through active quantum error correction [ ]. We may istinguish two limiting cases for moeling ecoherence in qubits. The first limit is that of inepenent qubit ecoherence. Here, each qubit couples inepenently to its own environment, without any interactions between these environments. For example, this may be the case if the qubits are spatially well-separate (relative to the typical coherence length of the environment) an only couple to their immeiate surrounings. Then the error processes affecting the qubits will be completely uncorrelate. Thus, if the probability of a particular error to affect one qubit is p, the probability of this error to occur in K qubits will be p K. Many error-correcting schemes are only efficient in correcting such single-qubit errors, an thus the assumption of inepenent ecoherence frequently unerlies these schemes. This assumption, however, is unrealistic when the qubits are locate spatially close to each other. In this case, all qubits approximately feel the same environment, an it is likely that errors will become correlate among multiple qubits. The limiting case corresponing to this situation is that of collective qubit ecoherence, in which all qubits couple to exactly the same environment. A. Correction of ecoherence-inuce quantum errors Consier a single qubit S, initially escribe by a pure state ψ an interacting with an environment E. One can show that an arbitrary evolution of the combine qubit environment state can always be written in the form ψ e I ψ e I + (σ s ψ ) e s, (62) s=x,y,z where the Pauli operators σ s act on the Hilbert space of S, an e I an { e s } are environmental states that are not necessarily orthogonal or normalize. Thus, any influence of the environment on the qubit can be expresse simply in terms of a weighte sum of the Pauli operators an the ientity operator acting on the original state of the qubit. The effects of σ x an σ z on the qubit state are

15 15 often referre as a bit-flip error an phase-flip error, respectively. If we restrict our attention to environmental entanglement an the resulting ecoherence effects, then only phase-flip errors nee to be taken into account. For N qubits, Eq. (62) generalizes to ψ e i (E i ψ ) e i. (63) Here ψ is the initial N-qubit state, an the error operators E i are tensor proucts of N operators involving ientity an Pauli operators. Equation (63) represents a worst-case scenario. In many cases, simplifie versions can be use. One important case is that of partial ecoherence. Here, only a small number K < N of qubits become entangle with the environment between two successive applications of an error-correcting mechanism. Then it will be sufficient to restrict our attention to the 2 K possible error operators mae up of at most K operators σ z an N K ientity operators. In the case of inepenent qubit ecoherence, we only nee to consier a collection of inepenent phase-flip errors acting on single qubits, represente by error operators of the form E = I I σ z I I. Given the entangle state on the right-han sie of Eq. (63), the goal of quantum error correction is to restore the initial (unknown) state ψ. We let an ancilla, escribe by an initial state a, interact with the qubit system such that [ ] a (E i ψ ) e i a i (E i ψ ) e i. (64) i i Let us assume that the ancilla states a i are at least approximately mutually orthogonal, such that they can be istinguishe by measurement. We now measure the observable O A = i a i a i a i on the ancilla, with a i a j for i j. The projective measurement will yiel a particular outcome, say, a k, an lea to the reuction of the entangle state, a i (E i ψ ) e i a k (E k ψ ) e k. (65) i The outcome a k of the measurement tells us the countertransformation neee to restore the initial qubit state. Applying E 1 k = E k to the system gives a k (E k ψ ) e k E 1 k a k ψ e k. (66) Note that, as require in orer to avoi introucing aitional ecoherence in the computational basis of the qubit system, we have obtaine no information whatsoever about the state of the system. This account of quantum error correction has been highly iealize. Let us mention three complications. First, it is impossible to esign an interaction between the computational qubits an the ancilla that woul allow us to istinguish, by measuring the ancilla, between all possible errors. Secon, in realistic settings the error operators E i may be very complex, an it remains to be seen whether an how the corresponing countertransformations can be applie without introucing significant aitional ecoherence. Thir, the ancilla qubits are physically similar to the computational qubits an can therefore be expecte to be equally prone to environmental interactions (an thus ecoherence) as the computational qubits themselves. Since the inclusion of ancilla qubits increases the total number of qubits in the quantum computer, an since ecoherence rates typically scale exponentially with the size of the system, it will require sophisticate experimental esigns to ensure not only that quantum error correction works in practice, but also that it oes not aggravate the problem of qubit ecoherence. B. Quantum computation on ecoherence-free subspaces We introuce the concept of ecoherence-free subspaces (DFS) [48 57], or pointer subspaces [3], in Sec. II C 2. DFS allow us to encoe quantum information in quiet corners of the Hilbert space to protect it from environmental effects. In contrast with quantum error correction, DFS prevent errors from happening in the first place an thus represent a strategy for intrinsic error avoiance. The two limiting cases of inepenent qubit ecoherence an collective qubit ecoherence elineate the limits on the size of a DFS. To illustrate this relationship, let us consier the case of collective ecoherence of an N-qubit system interacting with an oscillator bath [48, 5, 52, 55, 132]. The interaction Hamiltonian for this generalize spin boson moel is taken to be [compare Eq. (56)] N H int = σ z (i) ) N (g ij a j + g ija j σ z (i) E i. i=1 j i=1 (67) The assumption of collective ecoherence implies that the couplings g ij (an thus the environment operators E i ) must be inepenent of the inex i. Then Eq. (67) becomes ( ) H int = E S z E. (68) i σ (i) z Recall that a DFS is spanne by a egenerate set of eigenstates of the system operators S α of the interaction Hamiltonian [see Eq. (2)]. Thus, in our case the DFS will be spanne by egenerate eigenstates of the collective spin operator S z. Any N-qubit prouct state of the computational basis states an 1 (the eigenstates of σ z with eigenvalues +1 an 1, respectively) will be an eigenstate of S z. There are 2N +1 ifferent possible integer eigenvalues m, ranging from m = N (corresponing

16 16 to the basis state 1 1 ) to m = +N (corresponing to ). The largest number of mutually orthogonal computational-basis states with the same eigenvalue m of S z is given by the set S of basis states with m =, i.e., those with N/2 qubits in the state. There are n = ( N N/2) such states in this set, spanning a DFS of imension n. For large values of N, we can approximate the binomial coefficient using Stirling s formula, ( ) N log 2 N 1 N/2 2 log 2(πN/2) N 1 N. (69) Therefore, in the limiting case of collective ecoherence, the imension of our DFS approaches the imension of the original Hilbert space, an the encoing efficiency approaches unity. For example, for N = 4 qubits, the set S = { 11, 11, 11, 11, 11, 11 } (7) spans a maximum-size DFS of imension six, to be compare with the imension of the original Hilbert space, which is 2 4 = 16. Thus, given the moel for collective ecoherence consiere here, using four physical qubits we can encoe up to two logical qubits in a DFS (since encoing three logical qubits woul alreay require a DFS of imension 2 3 = 8). As mentione in Sec. II C 2, the existence of a DFS correspons to a ynamical symmetry. Our moel represents a case of perfect ynamical symmetry, since the system environment interaction, Eq. (68), is completely symmetric with respect to any permutations of the qubits, thereby leaing to a DFS of maximum size. What happens if the symmetry is broken by aitional small inepenent coupling terms? It has been shown [49, 133] that, to first orer in the perturbation strength, the storage of quantum information in DFS is stable to such perturbations to all orers in time, but that the processing of such quantum information encoe in DFS is robust only to first orer in time. In the case of purely inepenent qubit ecoherence, the environment operators E i appearing in Eq. (67) will now iffer from one another. To fin a DFS, we follow the usual strategy [see Eq. (2)] of etermining a set of orthonormal basis states { s i } such that [ I (1) I (j 1) σ (j) z I (j+1) I (N)] s i = λ (j) s i (71) for all i an 1 j N. The only state fulfilling this eigenvalue problem is. Since we nee at least a two-imensional subspace to encoe a single logical qubit, the case of inepenent ecoherence in the spin boson moel oes not allow for the existence of a DFS for quantum computation. In the language of pointer subspaces, there is only a single exact pointer state, an this environment-superselecte preferre state of the system will be the groun state. In realistic settings, neither the assumption of purely inepenent ecoherence nor the limit of entirely collective ecoherence will be entirely appropriate. We can, however, use a DFS to protect the qubits from collective ecoherence effects, an we can recover from single-qubit errors ue to inepenent ecoherence using active errorcorrection methos. These two approaches can be concatenate [53] to enable universal fault-tolerant quantum computation even when the restriction to single-qubit errors is roppe [54, 134]. C. Environment engineering an ynamical ecoupling For reasonably large DFS to exist, the system environment interaction must exhibit a sufficiently high egree of symmetry. Such symmetries are unlikely to arise naturally in typical experimental settings. One way of overcoming this limitation is base on environment engineering. Here, one tries to generate certain symmetries in the structure of the system environment interactions. For example, an appropriately engineere symmetrization coul make superposition states in Bose Einstein conensates correspon to (approximate) egenerate eigenstates of the interaction Hamiltonian, in which case such states woul lie within a DFS, thereby significantly enhancing their longevity [135]. In ion traps, changing the parameters in the effective interaction Hamiltonian for the trappe ion allows one to select ifferent pointer subspaces an thereby control into which DFS the trappe ion is riven [76, 78, 136, 137]. Another approach to the active creation of DFS is known as ynamical ecoupling [ ]. Here timeepenent moifications are introuce into the Hamiltonian of the system that counteract the influence of the environment. These moifications take the form of sequences of rapi projective measurements or strong control-fiel pulses acting on the system ( quantum bang-bang control [138]). Even if the structure of the system environment interaction Hamiltonian is not known, ecoherence can be suppresse arbitrarily well in the limit of an infinitely fast rate of the ecoupling control fiel, thus ynamically creating a DFS (which then represents a ynamically ecouple subspace). In the realistic case of a finite control rate, sufficient (albeit imperfect) protection from ecoherence can be achieve via this ecoupling technique, provie the control rate is larger than the fastest timescale set by the rate of formation of environmental entanglement. VI. EXPERIMENTAL STUDIES OF DECOHERENCE Decoherence, of course, happens all aroun us, an in this sense its consequences are reaily observe. But what we woul like to o is to be able to experimen-

17 17 tally stuy the graual an controlle action of ecoherence. In this eneavor, several obstacles have to be overcome. We nee to prepare the system in a superposition of mesoscopically or even macroscopically istinguishable states with a sufficiently long ecoherence time such that the graual action of ecoherence can be resolve. We must be able to monitor ecoherence without introucing a significant amount of aitional, unwante ecoherence. We woul also like to have sufficient control over the environment so we can tune the strength an form of its interaction with the system. Starting in the mi-199s, several such experiments have been performe, for example, using cavity QED [11], mesoscopic molecules [144], an superconucting systems such as SQUIDs an Cooper-pair boxes [12]. Bose Einstein conensates [145] an quantum nanomechanical systems [146, 147] are promising caniates for future experimental tests of ecoherence. These experiments are important for several reasons. They are impressive emonstrations of the possibility of generating nonclassical quantum states in mesoscopic an macroscopic systems. They show that the quantum classical bounary is smooth an can be shifte by varying the relevant experimental parameters. They allow us to test an improve ecoherence moels, an they help us esign evices for quantum information processing that are goo at evaing the etrimental influence of the environment. Finally, such experiments may be use to test quantum mechanics itself [12]. Such tests require sufficient shieling of the system from ecoherence so that an observe (full or partial) collapse of the wavefunction coul be unambigously attribute to some novel nonunitary mechanism in nature, such as those propose in ynamical reuction moels [148 15]. This shieling, however, is ifficult to implement in practice, because the large number of particles require for the reuction mechanism to become effective will also lea to strong ecoherence [19, 151]. The superpositions realize in current experiments are still not sufficiently macroscopic to rule out collapse theories, although it has been emonstrate [113] that matter-wave interferometry with large molecular clusters (in the mass range between 1 6 an 1 8 amu) woul be able to test the collapse theories propose in Refs. [149, 15]; such experiments may soon become technologically feasible [1]. A. Atoms in a cavity In 1996 Brune et al. generate a superposition of raiation fiels with classically istinguishable phases involving several photons [11, 145, 152]. This experiment was the first to realize a mesoscopic Schröinger-cat state an allowe for the controlle observation an manipulation of its ecoherence. A rubiium atom is prepare in a superposition of energy eigenstates g an e corresponing to two circular Ryberg states. The atom enters a cavity C containing a raiation fiel containing a few photons. If the atom is in the state g, the fiel remains unchange, whereas if it is in the state e, the coherent state α of the fiel unergoes a phase shift φ, α e iφ α ; the experiment achieve φ π. An initial superposition of the atom is therefore amplifie into an entangle atom fiel state of the form 1 2 ( g α + e α ). The atom then passes through an aitional cavity, further transforming the superposition. Finally, the energy state of the atom is measure. This isentangles the atom an the fiel an leaves the latter in a superposition of the mesoscopically istinct states α an α. To monitor the ecoherence of this superposition, a secon rubiium atom is sent through the apparatus. After interacting with the fiel superposition state in the cavity C, the atom will always be foun in the same energy state as the first atom if the superposition has not been ecohere. This correlation rapily ecays with increasing ecoherence. Thus, by recoring the measurement correlation as a function of the wait time τ between sening the first an secon atom through the apparatus, the ecoherence of the fiel state can be monitore. Experimental results were in excellent agreement with theoretical preictions [153, 154]. It was foun that ecoherence became faster as the phase shift φ an the mean number n = α 2 of photons in the cavity C was increase. Both results are expecte, since an increase in φ an n means that the components in the superposition become more istinguishable. Recent experiments have realize superposition states involving several tens of photons [155] an have monitore the graual ecoherence of such states [156]. B. Matter-wave interferometry In these experiments (see Ref. [1] for a review), spatial interference patterns are emonstrate for mesoscopic molecules ranging from fullerenes [157] to molecular clusters involving hunres of atoms, with a total size of up to 6 Å an masses of several thousan amu (see Fig. 2) [158, 159]. Since the e Broglie wavelength of such molecules is on the orer of picometers, stanar ouble-slit interferometry is out of reach. Instea, the experiments make use of the Talbot Lau effect, an interference phenomenon in which a plane wave incient on a iffraction grating creates an image of the grating at multiples of a istance L behin the grating. In the experiment, the molecular ensity (at a macroscopic istance L from the grating) is scanne along the irection perpenicular to the molecular beam. An oscillatory ensity pattern (corresponing to the image of the slits in the grating) is observe, confirming the existence of coherence an interference between the ifferent paths of each iniviual molecule passing through the grating. Recent experiments have use an improve version of the original Talbot Lau setup [16], as well as optical ionization gratings [161].

18 NATURE COMMUNICATIONS DOI: 1.138/ncomms sics, single-particle regare as a parafeature of quantum objects of our macinciple has become g fiel of quantum h in many laboraerstaning of the antum systems an the observation of interference with successful experiur stuy focuses on on of the molecule e. We o this with ie useful molecu- 1 compares the size an PFNS1, with raphenylporphyrin F84 an TPPF152. olecules in a threepitza-dirac-talbot- rate in a thermal avitational free-fall eter itself consists mber at a pressure brane with 9-nm nm. Each slit of G 1 cular position that, s to a momentum elocalization an increasing istance er light wave with a en the electric laser creates a sinusoial matter waves. The such that quantum c molecular ensity tructure is sample al to G 1 ) across the of the transmitte MS). ae various techo liqui samples, a ial to maintain the we us to increase an many optimiere neee to meet with very massive um interferograms e 3. In all cases the e of the sinusoial l, excees the maxia significant multishown for TPPF84 interference conth iniviual scans an V obs = 49% for, we have observe an V obs = 16 2% Figure 1 Gallery of molecules use in our interference stuy. (a) The fullerene C 6 (m = 72 AMU, 6 atoms) serves as a size reference an 3 for calibration purposes; (b) The perfluoroalkylate nanosphere PFNS8 (C 6 [C 12 F 25 ] 8, m = 5,672 AMU, 356 atoms) is a carbon cage with eight perfluoroalkyl 2chains. (c) PFNS1 (C 6 [C 12 F 25 ] 1, m = 6,91 AMU, 43 atoms) has ten sie chains an is the most massive particle in the set. () A single tetraphenylporphyrin TPP (C 44 H 3 N 4, m = 614 AMU, 78 atoms) is the basis for the two erivatives (e) TPPF84 (C 84 H 26 F 84 N 4 S 4, m = 2,814 AMU, 1 22 atoms) an (f) TPPF152 (C 168 H 94 F 152 O 8 N 4 S 4, m = 5,31 AMU, 8 43 atoms). In its unfole configuration, the latter is the largest molecule in the set. Measure by the number of atoms, TPPF152 6 an PFNS1 are equally complex. All molecules are isplaye to scale. The scale bar correspons to 1 Å. NATURE COMMUNICATIONS 2:263 DOI: 1.138/ncomms Macmillan Publishers Limite. All rights reserve. y visibility (%) X Detector pressure (in 1 6 mbar) FIG. 2. Left: Z Molecular clusters G 3 use in recent interference experiments, rawn to scale (the scale bar represents 1 Å). G 2 Figure from Ref. [158]. (a) Fullerene C 6 (m = 72 amu, 6 atoms). (b) Perfluoroalkylate G 1 nanosphere S 3 PFNS8 (m = 5672 amu, 356 atoms). (c) PFNS1 (m = 691 amu, 43 atoms). () Tetraphenylporphyrin TPP (m = 614 amu, 78 Lens atoms). (e) TPPF84 (m S= amu, 22 atoms). (f) TPPF152 (m = 531 amu, 43 atoms). Right: Visibility of interference fringes S 1 of C 7 fullerenes as a function of the pressure ofoven the backgroun gas. Measure Laser values (circles) agree well with the theoretical preiction (soli line) [3, 112, 162] escribing an exponential ecay of visibility with pressure. Figure aapte from Ref. [11]. Figure 2 Layout of the Kapitza-Dirac-Talbot-Lau (KDTL) interference experiment. The effusive source emits molecules that are velocity-selecte by the three elimiters S 1, S 2 an S 3. The KDTL interferometer is compose of two SiN x gratings G 1 an G 3, as well as the staning light wave G 2. The optical ipole force grating imprints a phase moulation (x) opt P/(v w y ) onto the matter wave. Here opt is the optical polarizability, P the laser power, v the molecular velocity an w y the laser beam waist perpenicular to the molecular beam. The molecules are etecte using electron impact ionization an quarupole mass spectrometry. Decoherence is measure as a ecrease of the visibility of for this TPPF152 pattern (see Figure (Fig. 3), 2). in which Theour controlle classical moel ecoherence preicts ue Vto class collisions = 1%. This supports with backgroun our claim of true gas quantum particles interference [11, for 111] all these complex molecules. an ue to emission of thermal raiation from heate The most massive molecules are also the slowest an therefore molecules the most [163] sensitive hasones been to external observe, perturbations. showingin aour smooth particle ecay of visibility in agreement with theoretical preictions [3, 112, 162]. These successes have le to speculations that one coul perform similar experiments using even larger particles such as proteins an viruses [11, 164] or carbonaceous aerosols [165]. Such experiments will be limite by collisional an thermal ecoherence an by noise ue to inertial forces an vibrations [11, 164, 165]. C. Superconucting systems Superconucting quantum interference evices (SQUIDs) an Cooper-pair boxes have important applications in quantum information processing. A SQUID consists of a ring of superconucting material interrupte by thin insulating barriers, the Josephson junctions (Fig. 3a). At sufficiently low temperatures, electrons of opposite spin conense into bosonic Cooper pairs. Quantum-mechanical tunneling of Cooper pairs through the junctions leas to the flow of a resistancefree supercurrent aroun the loop (Josephson effect), which creates a magnetic flux threaing the loop. The collective center-of-mass motion of a macroscopic number ( 1 9 ) of Cooper pairs can then be represente by a wave function labele by a single macroscopic variable, namely, the total trappe flux Φ through the loop. The two possible irections of the supercurrent efine a qubit with basis states {, }. By ajusting an external magnetic fiel, the SQUID can be biase such that the two lowest-lying energy eigenstates an 1 are equal-weight superpositions of the persistent-current states an. Such superposition states involving µa currents were first experimentally observe in 2 using spectroscopic measurements [166, 167]. Their ecoherence was subsequently measure using Ramsey interferometry [168], as follows. Two consecutive microwave pulses are applie to the system. During the elay time τ between the pulses, the system evolves freely. After application of the secon pulse, the system is left in a superposition of an, with the relative amplitues exhibiting an oscillatory epenence on τ. A series of measurements in the basis {, } over a range of elay times τ then allows one to trace out an oscillation of the occupation probabilities for an as a function of τ (Fig. 3b). The envelope of the oscillation is ampe as a consequence of ecoherence acting on the system uring the free evolution of uration τ. From the ecay of the envelope we can infer the ecoherence timescale; the original experiment gave 2 ns [168], while subsequent experiments have achieve ecoherence times of several µs [169]. Superpositions states an their ecoherence have also been observe in superconucting evices whose key variable is charge (or phase), instea of the flux variable use in SQUIDs. Such Cooper-pair boxes consist of a small superconucting islan onto which Cooper pairs can tunnel from a reservoir through a Josephson junction. Two ifferent charge states of the islan, iffering by at least one Cooper pair, efine the basis states. Coherent oscillations between such charge states were first observe in 1999 [17]. In 22, Vion et al. [171] reporte thousans of coherent oscillations with a ecoherence time of.5 µs. Similar results have been obtaine for phase qubits [172, 173], emonstrating ecoherence times of several µs.