Einselection from Incompatible Decoherence Channels

Einselection from incompatible decoherence channels

Alexandre Feller,1, ∗ Guillaume Coeuret Cauquil,1 and Benjamin Roussel1, † 1Advanced Concepts Team, European Space Agency, Noordwijk, 2201 AZ, Netherlands Decoherence of quantum systems from entanglement with an unmonitored environment is to date the most compelling explanation of the emergence of a classical picture from a quantum world. While it is well understood for a single Lindblad operator, the role in the einselection process of a complex system-environment interaction remains to be clariﬁed. In this paper, we analyze an open quantum dynamics inspired by CQED experiments with two non-commuting Lindblad operators modeling decoherence in the number basis and dissipative decoherence in the coherent state basis. We study and solve exactly the problem using quantum trajectories and phase-space techniques. The einselection optimization problem, which we consider to be about ﬁnding states minimizing the variation of some entanglement witness at a given energy, is studied numerically. We show that Fock states remain the most robust states to decoherence up to a critical coupling.

I. INTRODUCTION herence channels [12–15]. Interesting physical effects are coming from the subtle interplay between those incompatible channels. For instance, the physics of a quantum The strangeness of the quantum world comes from the magnetic impurity in a magnetic medium can be modeled principle of superposition and entanglement. Explaining as an open quantum system problem in which the envi- the absence of those characteristic features in the classical ronment probes all three Pauli matrices, instead of only world is the key point of the quantum-to-classical tran- one as in the standard spin-boson model [16, 17]. De- sition problem. The fairly recent theoretical and experi- pending on the relative values of the coupling constants, mental progresses have largely reshaped our understand- different classical regimes emerge at low energy with one ing of the emergence of a classical world from quantum channel dominating the others. What’s more, for some theory alone. Indeed, decoherence theory [1] has greatly fine tuned cases, the more interesting phenomena of de- clarified how quantum coherence is apparently lost for an coherence frustration occurs [17] where all channels con- observer through the entanglement of the system with a tribute to suppress the loss of coherence of the system at large unmonitored environment. At the same time, this all energy scales. This kind of subtleties in the einselec- interaction with the environment allows to understand tion process have also been studied in circuit QED [18– the emergence of specific classical pointer states, a pro- 22] where, once again, the emergent classical picture is cess called einselection. strongly dependent on the structure of the environment However, in the case of complex environments, the and the relative strength of its decoherence channels. physics of decoherence can be much more involved. It is then necessary to consider the structure of the envi- Thus, in the case of several incompatible decoherence ronment. A recent approach exploring this path, called channels, the problem of the emergence of a privileged quantum Darwinism [2,3], considers an environment basis is far from trivial. In this paper, we analyze the composed of elementary fragments accessing a partial einselection process of a system in contact with two com- information about the system. While a lot of insights peting decoherence channels inspired by cavity QED ex- were originally model-based [4–8], it has been proved periments. The system a mode of the electromagnetic that some basic assumptions of the quantum Darwinism field confined in a high-quality cavity. The two sources approach, like the existence of a common objective ob- of decoherence, one in the coherent state family and the servable, come directly from the theoretical framework of other in the number basis, comes for the imperfections quantum theory [9–11]. of this cavity and the atoms used to probe the field that arXiv:2001.10851v2 [quant-ph] 9 Apr 2020 Another possible source of complexity comes from the are sent through the cavity. We base our analysis on the possibility that the environment can measure different Lindblad equation to describe the effective open quan- observables of the system. Formally, the effective dynam- tum dynamics. The problem is solved exactly using the ics of the system will be described by many Lindblad op- quantum trajectory, the characteristic function and the erators which may be non commuting. When right eigen- quantum channel approaches allowing to properly char- vectors of a Lindblad operator exist, which correspond to acterize the different parameter regimes. These methods exact pointer states, we say that the operator describes allow to go beyond the formal solution [14, 15] by offering a decoherence channel. Non-commuting Lindblad oper- a clearer physical representation of the dynamics and the ators will give rise to what we call incompatible deco- relevant timescales of the the problem. Still, the einselection problem, which is about finding the most robust (approximate) states to the interaction with the environment, has to be solved numerically. We choose those ∗Electronic address: [email protected] states as the pure states that minimize the short-time †Electronic address: [email protected] variation of some entanglement witness (the purity) with 2

the environment at a given fixed energy. Depending on † bk, b the relative values of the coupling constant, we find the (a) k gk intuitive result that the einselected states interpolate be- χ(t) Atom Atom’s EM env (a) tween Fock and coherent states. However, we uncover ωeg ωk the remarkable fact that Fock states remain exactly the Cavity optimal einselected states up to a critical value of the (c) † g ck, c couplings that can be obtained analytically. Finally, in ωc k k the long-term dynamics, the einselection process appears Cavity’s EM env (c) to be much more complicated. ωk The paper is structured as follows. In SectionII, we present a summary of techniques to derive the reduced FIG. 1: Dynamics of the cavity. The cavity is coupled to dynamics from the full system-environment unitary evo- its own electromagnetic environment and to an atom, itself lution. This section can be skipped if the reader accepts coupled to its own electromagnetic environment. We suppose Eq. (4). The core results are presented in SectionIII that the two electromagnetic baths are not coupled. Even- where the model is solved exactly. The einselection pro- tually, the coupling of the cavity to its bath leads to photon cess is analyzed in SectionIV and we present numerical losses. The coupling with the atom, in the dispersive regime, evaluation of the Wigner function of the system which leads to decoherence on the photon basis. are used to properly analyze the different einselection regimes. We conclude in SectionV by discussing how our analysis could be generalized and understood from mode on the coherent state basis of the electromagnetic a more abstract perspective through the algebraic struc- field [26], and introduces dissipation. ture of the jump operators. We thus have two natural decoherence channels in this experimental setup, one which selects photon number states and the other coherent states. However, those two basis are incompatible in the sense that they are asso- II. MOTIVATIONS FROM CQED ciated to complementary operators, the phase and number operators. This strong incompatibility between those By managing to entangle a single mode of the electro- classical states motivates the question on which classical magnetic field and an atom, cavity quantum electrody- pictures, if any, emerge from such a constrained dynam- namics (CQED) is a nice experimental setup to explore ics. the foundations of quantum theory, quantum information Before analyzing this problem in details, let’s first put and the physics of the quantum to classical transition. the previous discussion on firmer ground by deriving an One implementation of CQED [23] uses an electromag- open quantum system dynamics of the Lindblad form. netic mode trapped in high quality factor mirrors as a We use a Born-Markov approximation of an effective system which is probed by a train of atoms acting as two- Hamiltonian describing a CQED experiment represented level systems (qubits) conveniently prepared. The setup on Fig.1. As a starting point, consider the following can work in different regimes where the qubit is in reso- Hamiltonian: nance or not with the field. The off-resonance functioning mode, also called the dispersive regime, is particularly in- H = HS + HE + HSE , (1a) teresting since the atoms can be thought as small trans- HS = ~ωc N, (1b) parent dielectric medium with respect to the field with 1 X (c) X (a) an index of refraction depending on its state. The atom H = ω σ + ω c† c + ω b† b E 2~ eg z ~ k k k ~ k k k is able to register some phase information about the field k k making it a small measurement device. In fact, a train X (a) † + σz · ~g (b + bk) , (1c) of atoms can be thought as a non-destructive measure- k k k ment device probing the number of photons in the field. X (c) † Focusing our attention on the field itself, the atoms have HSE = ~χ(t) N · σz + a · gk ck + h.c. , (1d) to be considered as part of the environment of the field k (even if it is well controlled by the experimentalist) giv- with (a, b, c) ladder operators of harmonic oscillators as- ing rise to a decoherence channel having the Fock states sociated to the system and two environments respec- as pointer states [23–25]. Similar experiments can also tively, N = a†a the number operator for the system oscil- now be performed on circuit QED platforms [22]. lator a and σz the z Pauli matrix. The system of interest However, a second source of decoherence exists, not is the mode confined in the cavity. Its free Hamiltonian controlled by the experimentalist this time, and has its HS is just the free harmonic oscillator at frequency ωc. origin in the imperfections of the cavity. Indeed, while The cavity mode is coupled to two other systems: the quality factor is high enough to see subtle quantum phenomenon, photons can still get lost over time in 1. The atom at frequency ωeg. It is coupled to the the remaining electromagnetic environment. This second cavity in the dispersive regime defined as a non decoherence channel leads to a decoherence of the field zero dephasing ∆ = ωeg − ωc between the qubit 3

p (c) (c) and the cavity. This is the first term of HSE. La(ω) = 2G (−ω) a with G the correlation func- The coupling constant χ(t) is given by g2(t)/4∆ tion of the environment field modes at zero temperature. where g(t) is the coupling constant in the Jaynes- Before going to the main analysis of the model, two Cummings model. This atom is subjected to an remarks have to be made. The first remark is that if we electromagnetic environment described by the har- were to prepare the experimental setup in the resonant (a) (a) regime ∆ = 0, the atom-field interaction is modified in monic modes (ωk , gk ) coupled to it by a spin- boson model. Both the atom and its electromag- such a way that the atom can emit or absorb a photon netic environment form the first decoherence chan- from the mode. Staying in the Markovian regime would nel. result in a dissipative dynamics equivalent to a photon emission and absorption albeit with very different transi- (c) tion rates. Thus, running the experiment in the resonant 2. Other harmonic modes ωk describing the electromagnetic environment of the cavity. In the secular regime will not result in the dynamics we are interested approximation, this environment leads to photon in. The second one refers to the system-environment cut losses in the cavity. This is the second decoherence and the Markovian hypothesis. Here, we chose to con- channel. sider as the system only the field mode, all the other degrees of freedom then being part of the Markovian en- The first decoherence channel is given by the atom vironment. However, as it was done in [18], it is also passing through the cavity. Its effect on the field is char- possible to consider as the system the field and the qubit acterized by a correlation function of the form: while all the other electromagnetic field modes form the Markovian environment. Different decoherence channels (a) 0 0 0 0 Gρ (t, t ) = hχ(t)σz(t) χ(t )σz(t )iρ = χ(t)χ(t ) . (2) acting on the qubit and/or on the field can be considered while the non-trivial internal Jaynes-Cummings dynam- We remark that, in this case, the correlation function ics adds another level on complexity. is independent of the state of the qubit ρ. We will rewrite the correlation function in terms of the vari- 0 (a) 0 able τ = t − t . Supposing that Gρ (t, t ) satisfies III. DISSIPATIVE DYNAMICS the usual assumptions of the Born-Markov approximation scheme (stationnarity and fast decay), the opera- A. Position of the problem tors appearing in the Born-Markov equation are of the +∞ (a) −iωτ form 0 G (t, τ) e dτ N. From there follows a The previous discussion shows that it is possible in Lindblad´ equation with the jump operator Ln(ω) = principle to engineer a complex open quantum dynamics p2G(a)(−ω)N. Unfortunately, it is not possible to sat- with two incompatible decoherence channels. Building on isfy those requirement for the function χ(t) (indeed, sta- this motivating example, the problem we are interested in tionarity implies that χ should be a phase but, being is to understand the einselection process of a field mode real, it must be a constant which cannot satisfy the de- modeled by a simple harmonic oscillator subject to two cay requirement). Thus, the time dependence must be decoherence channels: one is induced by the dispersive kept in full generality in this model, breaking the usual interaction with a train of atoms leading to decoherence stationarity assumption. This is not a problem for the on Fock states and the other is induced by loss of photons derivation of a master equation of the Lindblad form, in the cavity leading to decoherence on coherent states. the only difference being that the rates will be time de- Abstracting ourselves from the CQED context, we con- pendent. Still making the assumption of fast decay in sider from now on an open quantum dynamics modeled the time coordinate τ, the usual steps of the deriva- by a Lindblad equation with the Hamiltonian HS and tion can be followed. We then end up with a time- two time-independent jump operators: √ √ √ dependent Lindblad term proportional to the operator L = κ a L = κ a†a = κ N. (3) p (a) a a n n n N: Ln(t, ω) = 2G (t, −ω)N. A sufficient condition for the fast decay assumption Those two operators do not commute with each others to be valid is to have an exponential decay. This is the and therefore can’t be simultaneously diagonalised. The case in the Rydberg atoms experiment where the cou- incompatibility between decoherence channels we are re- pling constant follows the Gaussian beam in the cavity. ferring to has to be understood in this sense. Still, their If we send a stream of atoms, we still have to make sure commutator [Ln,La] ∝ La remains simple enough and that the assumption of fast decay in τ is valid. This im- this is the key to find an exact solution [14]. plies that the atoms must be sent as a group to have the Given the two quantum jumps La and Ln, the Lind- exponential decay of the correlation function. blad equation we want to analyze is: The second decoherence channel comes from the leaky 1 ∂ ρ = −iω [N, ρ] + κ NρN − {N 2, ρ} cavity and does not present any analytical difficulties. t c n 2 Supposing that those degrees of freedom are at equilib- † 1 rium and at zero temperature, the Born-Markov equation + κa aρa − {N, ρ} , (4) then reduces to a Lindblad term with the jump operator 2 4 where N is the number operator and ωc the frequency of If the initial state at t = 0 is a Fock state |n0i, only −κat the field mode. the isotropic contribution hN(t)i = n0 e survives. In To gain some intuition on the physics behind this dy- this case, the fluctuations are decreased solely because namics, let’s study the evolution of some average values. of the loss of energy in the cavity. This is due to the We will focus on the average position in phase space, fact that the La channel does not create any coherences which can be recovered directly from the average value between different Fock states, while the Ln channel does of the annihilation operator hai, as well as the variance not affect statistical mixtures of Fock states. around this position which are related to the average However, the situation is more subtle if we prepare photon number hNi and the square of the annihilation a coherent state. A coherent state |αi such that α = 2 √ iϕ 2 operator ha i. From those values, we can extract the av- n0e gives the values hN(0)i = n0 and ha (0)i = 2iϕ erage position and the fluctuations on an arbitrary axis n0e . The fluctuations for such a state are evolving xˆθ = cos θ xˆ + sin θ pˆ. according to: The observable N is in this case of particular interest. First of all, it gives access to the average energy of the 1 ∆x2(t) = + cavity. Second, it is not affected by the proper dynamics θ 2 of the cavity nor by the L jumps. A direct computa- n n e−κat(1 − e−κnt) 1 − e−κnt cos (2(ϕ − θ − ω t)) . tion gives the standard exponential decay of a damped 0 c (9) harmonic oscillator with rate κa:

hN(t)i = e−κathN(0)i . (5) As in the Fock state case, the fluctuations are exponentially suppressed in time as the cavity looses energy with This solutions gives the intuitive result that photon loss a rate κa. However, the situation is more interesting, be- induces a decrease of the average number of photons in cause of the competition between both channels on the the cavity. isotropic part of the fluctuations: on top of the exponen- On the contrary, both decoherence channels affect the tial suppression, the Ln channel tends to increase fluctua- average values of the annihilation operator a and its tions up to the maximum possible for the average energy square: still stored in the cavity. As we will see later, this comes from the fact that the coherences between different pho- −iωct −κnt/2 −κat/2 ha(t)i = e e e ha(0)i , (6) ton numbers are suppressed over a timescale 1/κn. ha2(t)i = e−2iωcte−2κnte−κat ha2(0)i . (7) The anisotropic part of the fluctuations, in this case, is always smaller than the isotropic part over the vacuum. We see that the average position in phase space oscillates The fluctuations will thus stay always above the vacuum at the frequency of the cavity and is exponentially sup- fluctuations in any directions: this dynamics does not pressed at a rate (κa+κn)/2, different than the decay rate exhibit squeezing. At short times, however, the fluctua- 2 2 of the average energy. We expect that, for some states, tions behave as ∆xθ(t) ' 1/2 + 2n0κnt sin (ϕ − θ − ωct). this separation of timescales will lead to an increase of They stay identical in the axis given by (0, α), but are fluctuations at intermediate times. increased with a rate n0κn in the orthogonal direction. To put this statement on firmer ground, it is instruc- At longer times, the anisotropic fluctuations decay, with tive to compute the average fluctuations along an axis xθ an exponential rate (κa + κn), faster than the isotropic making an angle θ with the horizontal in phase space. A decay rate κa. direct computation gives: Those timescales are very important to properly follow the einselection process and we will analyze them in detail 1 ∆x2(t) = + hN(t)i − |ha(t)i|2 after having obtained the general solution. θ 2 + Re ha2(t)i − ha(t)i2 e−2iθ . (8)

The first term 1/2 remains even when the cavity is empty B. Quantum trajectory approach and corresponds to the fluctuations of the vacuum. It is also the minimal isotropic fluctuations that satisfy the The Lindblad equation (4) can be solved exactly from Heisenberg principle. The second term corresponds to the quantum trajectory approach [27]. One of the mo- isotropic fluctuations above the vacuum state and are, for tivations for using this approach is that it is now pos- instance, zero for coherent states but not for a thermal sible to explore the dynamics of well-controlled quan- density matrix. Finally, the last term is the anisotropic tum systems at the level of a single experimental real- part of the fluctuations and can be either positive or neg- ization [28]. In this sense, it is closer to the latest experi- ative. Even though the overall fluctuations must satisfy ments studying decoherence. The strategy will be to first Heisenberg inequalities, states can exhibit fluctuations write the stochastic Schrödinger equation for the system, below the vacuum ones in some directions. This prop- parametrise properly a trajectory, write its relative state erty is called squeezing and is of particular interest for and finally average over all possible trajectories to obtain quantum technologies. the reduced density matrix. 5

The stochastic Schrödinger equation associated to the all the phases accumulated between each La jumps eas- Lindblad Eq. (4) can be written as: ily and then average over all La jump events. For the initial Fock state |mi, only the diagonal elements of the |ψc(t + dt, [µ])i = reduced density matrix can be non zero. No coherences  are induced since a Fock state remains a Fock state and 1 − dt iωa†a + κa a†a + κn (a†a)2 |ψ (t, [µ])i  2 2 c no superposition appears. This immensely simplifying  if µ(t) = 0, √ feature comes directly from the special commutation re- † −i dtκn a a |ψc(t, [µ])i if µ(t) = 1, lation of the jump operators. We obtain:  √ −i dtκa a |ψc(t, [µ])i if µ(t) = 2. (10) m ρ = (1 − e−κat)ke−κat(m−k) . (12) m−k,m−k k where µ(t) takes the values 0, 1, and 2 if there is no quantum jump, a jump La or a jump Ln respectively. A trajectory will be then parametrized by the type of jump and Starting from a Fock state |mi, the state |m − ki is reached if k photons have leaked. The associated prob- their occurrence time. We stress that the state |ψc(t, [µ])i m −κat k −κat(m−k) is not normalized, the probability of the trajectory t 7→ ability is given by p(t, k) = k (1 − e ) e , which is the classical probability for a binomial experi- µ(t) being given by p[µ] = hψc(t, [µ])|ψc(t, [µ])i. ence B(m, p), where the probability p for a photon to leak To have a better intuition of how a state evolves, let’s −κ t start from a Fock state |mi. Since it is an eigenstate of between 0 and t is p = 1−e a . We recover the standard a†a, the trajectories µ = 0 and µ = 1 do not change the evolution of a damped harmonic oscillator prepared in a state and only induce a phase. However, the jump µ = Fock state. 2 induces a photon loss and the state is changed into The coupling constant κn does not appear in this ex- |m − 1i with a phase. Thus, a proper parametrization pression which means that the dephasing induced by an of a trajectory is to slice the evolution according to the atom flux has no effect on a Fock state. Indeed, we saw La jumps, as shown in Fig.2. There, a trajectory is that Ln jumps induce a dephasing between Fock states. parametrized by: Since a unique trajectory keeps the cavity in a single Fock state, this dephasing is only a global phase shift and thus • Na jumps La, indexed by the letter σ, occurring at has no effect at all. times τ , σ Since Fock states form a basis of the Hilbert space of • Nσ jumps Ln in the slice σ, indexed by the letter the system, an analogous computation starting from any s, occurring at times tσ,s ∈]τσ, τσ+1[. superposition of Fock states gives the general solution:

m+n κn 2 −iω(m−n)t −κa t − (m−n) t |mi |m − 1i |m − k + 1i |m − ki ρm,n(t) = e e 2 e 2 ··· s X m + Na n + Na 0 τ1 τ2 τk−1 τk t × (1 − e−κat)Na Na Na Na FIG. 2: Parametrization of a quantum trajectory build from √ √ † × ρm+Na,n+Na (0) . (13) two types of quantum jumps La = κa a and Ln = κn a a. The times τσ correspond to the La jumps and the quantum state between each jump is above each slice. The physical content of this general exact expression is more transparent if we initially prepare a superposition For a given slice σ ∈ [[1, k + 1]] where τk+1 = t and of coherent states. What’s more, those are the typical τ0 = 0, the relative state obtained from Eq. (10) is pro- quantum states that are used in cavity quantum electro- portional to the Fock state |m − σi. By denoting α(t, [µ]) dynamics to study the decoherence process. Let’s then |α i+|α i the proportionality constant, we ﬁnd: consider the state |Ψ i = +√ − where α = α e±iθ/2, c N ±

Nσ−1 α ∈ C and N is a normalization factor. The initial den- Y p sity matrix is then given by: |ψc(τσ, [µ])i = −i κndtσ−1,s(m − σ + 1) s=1 p 1 × −i κ dτ (m − σ + 1) ρm,n(0) = ρm,n(α+) + ρm,n(α−) a σ N 2 × e−(i(m−σ+1)ωc+(m−σ+1) κn/2+(m−σ+1)κa/2)(τσ −τσ−1) + ρm,n(α+, α−) + ρm,n(α−, α+) , (14)

× α(τσ−1, [µ]) |m − σi . (11) where ρm,n(α) = hm|αihα|ni refers to the matrix element Having the formal form of the state of the system rel- mn in the Fock basis of the density matrix of the coherent ative to a given quantum trajectory, we can obtain the state |αi. The last two terms correspond to interferences reduced density matrix by summing over all possible tra- between the two coherent states, with ρm,n(α+, α−) = jectories. Our parametrization is such that we can re-sum hm|α+ihα−|ni. Using Eq. (13), the time-evolved density 6 matrix at time t is given by: using that |α|2 = hni. Over time, this channel tends to empty the cavity. The relaxation timescale τr deduced −κ t/2 1 (m−n)2 from the equation α(t) = e a α is such that: −κn t ρ (t) = e 2 ρ (α (t)) + ρ (α (t)) m,n N m,n + m,n − 2 −κ t iθ 1 + e−|α| (1−e a )(1−e )ρ (α (t), α (t)) + h.c. , τ = . (19) m,n + − r κ (15) a

A second decoherence timescale τn is associated to the −iω t − κa t where α (t) = α e c e 2 . The effect of the two de- ± ± Ln channel. In the Fock basis, its expression is straight- coherence channels can be clearly identified. First, the forwardly given by τ = 2/κ (m − n)2. Still, we can 2 −κ t iθ s n term d (t) = e−|α| (1−e a )(1−e ) is the usual decoher- a √ rewrite it in a form more suitable for coherent states. ence factor coming from the jump La = κaa (damped Indeed, coherent states have a Poissonian distribution harmonic oscillator) which tends to destroy coherences of photons with an average hni = |α|2 and a variance between coherent states. On the other hand, the term ∆n = p|α|2 = phni. Thus, the characteristic width ∆n 2 −κ (m−n) t e n 2 tends to destroy coherences between Fock of a coherent state gives a characteristic upper bound states which are the natural pointer states of the channel m − n ≤ ∆n. This allows us to extract a characteristic √ Ln = κnN. time which, as we will see, correspond to a spreading in The interesting and surprising aspect of Eq. (15) is the angle variable in phase space: that, while the Lindblad jump operators do not commute, the overall evolution is, in a sense, decoupled. This can be 2 τs = . (20) made precise by using the quantum channels perspective. κnhni Indeed, by considering the superoperators H = −i(H¯ ⊗ 1 1 P ¯ 1 1 † 1 ¯† ¯ 1 This timescale corresponds to the short time effect of − ⊗H) and D = µ Lµ ⊗Lµ − 2 ⊗LµLµ − 2 LµLµ ⊗ (the bar notation corresponds to complex conjugation), Ln. The long time effect is given for m − n = 1 and the Lindbald equation can be formally integrated as [29]: corresponds to a disappearance of the Fock states superposition. The associated timescale τc, which corresponds to the formation of a rotation-invariant crown in phase ρ(t) = e(H+D)tρ(0) = L (t)ρ(0) . (16) H,[Lµ] space, is given from Eq. (15) by: The notation L (t) is here to remind us that the H,[Lµ] 1 quantum channel depends on the set of jump operators τc = . (21) L through the operator D. In our problem, we have two κn µ √ non-commuting jump operators L = κ a and L = √ a a n Thus, the two natural decoherence timescales for coher- κ N. Astonishingly, their respective quantum channels n ent states are τ and τ , corresponding to the loss of do commute, leading to: a c coherence on the coherent state and number state basis respectively. The scaling τ /τ ' κ /κ hni implies that L (t) = L (t)L (t) = L (t)L (t) . a c n a H,La,LN H,La LN LN H,La for sufficiently high energy, we can have a separation of (17) decoherence times, by first seeing a decoherence over the Thus, while the generators of the open quantum dynam- coherent states followed by one on the number basis. Ta- ics do not commute, the associated quantum channels do bleI summarizes the different timescales, their expres- (and decouple in this sense). The full dynamics of the sion and interpretation when a superposition of coherent model can be solved by parts by looking at the evolution states is initially prepared. However, as we will see over of both quantum channels separately, which is an easy the next sections, the interpretation of those timescales task. Nevertheless, this does not trivialize the einselec- is relative to the class of pointer states we use like, for tion problem of finding the exact or approximate pointer instance, when we initially prepare Fock states. states that entangle the least with the environment. The emergence of a classical picture does depend on the “fine structure” of the dynamics. C. Phase-space approach Two natural decoherence timescales can be defined, still considering the initial state to be the superposition 1. The general solution of coherent states |Ψci. The first one is the usual decoherence timescale τa of a superposition of coherent states Last section presented the general solution of the Lind- induced by the decoherence channel La. Its expression blad Eq. (4) using the quantum trajectories approach. It is directly given by the exponential modulation of the is also possible to solve the same problem from a phase- coherence term in the general solution Eq. (15): space perspective using characteristic functions of the density matrix. This approach offers interesting physi- 1 cal insights on the dynamical evolution imposed by the τa = 2 , (18) 2 hni κa sin (θ/2) two incompatible channels. 7

TABLE I: Comparison of the timescales relative to the classes of exact pointer states Timescale Expression Coherent state (C) Fock state (F)

2 Decoherence of coherence states τa = 1/2 hniκa sin (θ/2) Decoherence (C) Spreading Relaxation τr = 1/κa Relaxation Relaxation & Decoherence (C) Spreading in angle τs = 2/κnhni Spreading Decoherence (F) Crown formation τc = 1/κn Decoherence (F)

Given a density operator ρ, we associate a function By applying the method of characteristics, solving the ∗ Cρ(λ, λ ), with λ ∈ C, defined as: equationsr ˙ = κa andn ˙ = 0, we obtain the general solution for an initial condition C0(r0, n0): ∗ λb† −λ∗b Cρ(λ, λ ) = tr ρe e . (22) −κ t/2 iω n− κn n2 t C (r, n, t) = C (re a , n) e( c 2 ) . (26) This function is called the characteristic function adapted ρ 0 to the normal order. All the normally-ordered average From the radial damping, we recover the usual damping values can be recovered from it. It is an integral trans- rate leading to decoherence on the coherent state basis. form of the Wigner function that we will define and use Besides this usual behavior, we have a new damping term extensively latter on. Now, by taking each term of the depending on the square of the Fock variable n which Lindblad Eq. (4), it is a straightforward computation, leads to the decoherence in the Fock basis with the char- using the BCH formula, to write it in terms of the characteristic angle-spreading timescale τ already uncovered acteristic function as: s in SectionIIIB. κa ∗ κa ∂ C + λ − iω ∂ C + λ + iω ∂ ∗ C t ρ 2 c λ ρ 2 c λ ρ κn ∗ 2 = − (λ∂ − λ ∂ ∗ ) C . (23) 2. Fock state decoherence 2 λ λ ρ By using polar coordinates λ = r eiθ, we obtain the more Different initial states can be prepared. We will natu- transparent form: rally focus on two classes of states, Fock states |ni and κ κ coherent states |αi. As a warm up example, suppose ∂ + a r∂ − ω ∂ C (r, θ, t) = n ∂2C (r, θ, t) . t 2 r c θ ρ 2 θ ρ that only the channel Ln is present. If we prepare a (24) Fock state |ni, whose characteristic function is given by 2 C|ni(λ) = Ln(|λ| ) where Ln is the Laguerre polyno- This equation is of the Fokker-Planck type with the right- mial of order n, we know that it will not be affected by hand side being a diffusion term with a diffusion coeffi- the environment and will evolve freely as can be read- cient given by the coupling constant κn. Thus, the chan- ily checked from Eq. (26). The important remark here nel Ln tends to spread in angle the characteristic func- is that the phase-space representation of a Fock state or tion. This is in accord with the first discussion of the any statistical mixture of them is rotation invariant. evolution of fluctuations given through the average val- If we now prepare a quantum superposition of Fock ues. Note that we could even consider an environment √ states (|n0i + |n1i)/ 2, it is straighforward to show at finite temperature and obtain a similar equation (with that the coherence part of the characteristic function an inhomogeneous term) that can also be solved exactly. ∗ λb† −λ∗b C01(λ, λ ) = tr |n0ihn1| e e evolves as: When only the channel La is present, the differential equation is first order and can be solved by the method κn 2 ∗ − (n1−n0) t of characteristics [30, 31]. This is not directly the case C01(λ, λ ) = e 2 C01(r, (ωc − θ)t) . here but can be remedied by going to the Fourier domain with respect to the variable θ. Doing this is indeed quite The coherence term between Fock states are damped intuitive if we remember that the conjugated observable by an exponential factor with a characteristic timescale 2 associated to θ is the number operator N which is the 2/(n0−n1) κn scaling as the quadratic inverse of the“dis- natural observable of the problem, and that both are re- tance” between the two components of the state. The is lated by a Fourier transform. the expected decoherence dynamics for the exact pointer Let’s then define a new characteristic function states of a decoherence channel. inθ Cρ(r, n) = Cρ(r, θ) e dθ which is the Fourier series of Cρ(r, θ).´ We then end up with an in-homogeneous first-order partial differential equation: 3. Coherent states and the Wigner representation κa 2 κn ∂t + r∂r Cρ(r, n, t) = iωcn − n Cρ(r, n, t) . 2 2 In the spirit of cavity quantum electrodynamics ex- (25) periments, it is more appropriate to study the evolution 8 of coherent states and their superposition. Before discussing the dynamics of such states, it is necessary to in- -0.6 -0.3 0.0 0.3 0.6 troduce a better-suited phase-space representation then the one defined by Eq. (22). Indeed, this latter func- κnt = 0 κnt = 1/20 κnt = 2 tion is a complex-valued function which is not the best choice for representation purposes. From the character- 5 ∗ ) istic function Cρ(λ, λ ), we can recover an equivalent, α ( 0

real-valued, phase-space representation called the Wigner = function Wρ(α) of the state ρ (with the parameter α ∈ C) -5 deﬁned as the following integral transform [30, 32]: -5 0 5 -5 0 5 -5 0 5 1 2 ∗ ∗ W (α) = C (λ) e−|λ| /2 eαλ −α λ d2λ . (27) <(α) ρ π2 ˆ ρ

Using the position and momentum coordinates in phase FIG. 3: Evolution of a superposition of coherent states from space α = x + ip, we recover the common expression: t = 0, subject to the Ln channel only (κa = 0) in the Wigner function representation. The Gaussian spots are spread over −iup du a timescale√ given by κnτs = 2/hni. For the chosen parame- Wρ(x, p) = hx + u/2|ρ|x − u/2i e , (28) ˆ 2π ters α = 40, it is given by κnτs = 1/20. At longer times κnt = 2, the superposition evolves toward a statistical mix- where the states |xi are eigenstates of the position oper- ture of Fock states with the characteristic rotation invariance ator x ∝ a + a†. The Wigner function is used in a wider in phase space. context than quantum optics [33, 34] and possesses a nice set of properties to represent quantum interferences in a transparent way. We will use it throughout this paper given by the convolution of the initial characteristic√ func- −θ2/2σ2 2 to represent our analytical and numerical results. Fur- tion with a Gaussian function Nσ(θ) = e / 2πσ thermore, since the proper dynamics of the field mode of the angle variable θ spreading in time with a variance 2 can be factored out, the dynamics is pictured without σ = κnt (see AppendixA1). In terms of the Wigner the rotation in phase space that it introduces. function, we have: Both for the numerical calculations as well as for un- +∞ derstanding the dynamics in phase space, it is helpful to 0 0 0 2 W (r, θ, t) = W0(r, θ ) Nσ =κnt (θ − θ ) dθ . (32) write the Wigner function in terms of its Fourier com- ˆ−∞ ponents of the polar angle θ. Thus, let’s decompose the Wigner function on the Fock state basis as: with W0 the initial Wigner function. This form displays clearly the diffusive dynamics induced by the L deco- X n W (x, p) = ρmnWmn , (29) herence channel in the phase angle, putting the initial m,n intuition on firm ground. dχ Figure3 represents the evolution of the Wigner func- W (x, p) = hx + χ/2 | mi hn | x − χ/2i e−ipχ . mn ˆ 2π tion of a superposition of coherent states. At a time scale (30) τs = 2/κnhni, the Gaussian spots start to spread along circles with a radius given by their respective amplitude. If we denote x + ip = reiθ, we find that for l ≥ 0: The same can be said for the Gaussian interference spot which is centered at the mid-point in phase space. At s (−1)k k! later times t & τc, the coherent states spread uniformly W (r, θ) = 2l e−ilθrkLl (2r2) , (31) and are completely decohered as a statistical mixture of k,k+l π (k + l)! k Fock states1.

k In fact, we can better understand the structure of the where Lm are generalized Laguerre polynomials. This Wigner function at ﬁnite times by explicitly writing the implies that the coherence given by ρk,k+l (k ∈ N, l ≥ 0) periodicity in the variable θ hidden in Eq. (32). Indeed, and ρk−l,k (k ∈ N, l ≤ 0) corresponds to the lth harmonics of the angle variable in the Wigner function. Notably, since the Ln channel acts by multiplying all the elements situated at a distance l from the diagonal 1 This aﬃrmation is true because of the rotation invariance of the by the same quantity, it corresponds to reduce the dif- Wigner function. Indeed, the Wigner function of a statistical ferent harmonics of the angle variable by an exponential mixture of Fock states is rotation invariant being a sum of func- 2 amount. This corresponds of the idea of a spread in the tion of the form Ln(r ). The converse is also true: from the fact angle variable. that the set of functions (Ylm(θ)Ln) form a basis of the functions f(θ, r), a rotation invariant Wigner function W (r) can be rep- If we now prepare a coherent state |αi whose character- resented as a sum of Laguerre polynomials. In terms of density α∗λ−αλ∗ istic function is given by C|αi(λ) = e , its evolu- operator (Fourier transforming the Wigner function), we end up tion in the presence of only the Ln decoherence channel is with a statistical mixture of Fock states. 9

-0.6 -0.3 0.0 0.3 La Ln

κnt = 0 κnt = 1/10 κnt = 2

5 t ∼ τa t ∼ τs ) α

( 0 = -5

-5 0 5 -5 0 5 -5 0 5 <(α)

t ∼ τr t ∼ τc FIG. 4:√ Depending on the√ preparation, with for instance here α+ = 40 and α− = − 10, the interference spot is totally washed out by the Ln channel (κa = 0) after the spreading timescale κnt = 1/10 leading at longer times κnt = 2 to a mixture of Fock states.

2π 0 0 we have W (r, θ, t) = W (r, θ0) ϑ θ−θ ; i κnt dθ 0 0 2π 2π 2π FIG. 5: Summary of different evolution and timescales of a with ϑ(z; τ) = P exp(´πin2τ +2πinz) the Jacobi theta superposition of coherent states under the effects of two deco- n∈Z θ κnt herence channels: the Ln channel induces a spreading of the function. When κnt 1, the expansion ϑ 2π ; i 2π ' n2 wavepacket while the La channel destroys interferences and P −κnt/2 1 + 2 ∗ e cos nθ can be used to approx- makes the system relaxes to the vacuum. n∈N imate the Wigner function. For instance, if κnt = 2, only the first harmonic of the Wigner function dominates with oscillations in amplitude of 74 % of the average One regime is when we have τa/τc ≈ κn/κahni 1 value, the second harmonic being less than 2 %, as can be or equally hni κn/κa. We expect in this case that seen on Figs.3 and4. This harmonic decomposition also Fock state decoherence dominates the dynamics. We can shows that in the presence of symmetries in angle, for in- refine our statement by unraveling two subregimes with stance W (r, θ +2π/p) = W (r, θ) for an integer p ≥ 2, the respect to the relaxation timescale τr: 2 first non-zero modulation term scales as e−κntp /2. Thus, • In the range τc τa, τr, or equally hni κn/κa symmetric initial states will decohere much quicker than and τr/τc ≈ κn/κa 1, we have a proper decoher- the ones that are not. ence on the Fock basis as we expected. Finally, note that the presence of an interference pat- tern on Fig.3 is not a signature of coherence between • However, for τr τc τa, or equally hni κn/κa Fock state. For a different preparation like the one on and τr/τc ≈ κn/κa 1, we end up in the almost Fig.4, the interference spot is completely washed out degenerate case hni 1 where we relax on the over a timescale of τs. vacuum state.

Thus, in the regime τa τc where the Fock decoherence IV. EINSELECTION PROCESS dominates initially, we see that the relaxation induced by the La channel forbids to transition toward a proper non- trivial coherent-state classical picture: before evolving Now that we have a proper analytic solution of the towards a proper statistical mixture of coherent states, problem and that we completely understand the evolu- the system relaxes to the vacuum. All in all, the emergent tion and characteristic timescales of each decoherence physically meaningful classical picture is given by Fock channel separately, which is summarized in Fig.5, we states. can analyze the einselection process in its full generality. From the einselection perspective, a richer dynamics can be found in the complementary regime where τa/τc ≈ A. Pointer states dynamics κn/κahni 1 or equally hni κn/κa. Figure6 shows the evolution of the Wigner function of a coherent state (with initially 40 photons on average) for this situation. Intuitively, we expect the classical picture that emerges We can at least see three different regimes: must be dominated by the quantum channel which has the strongest coupling constant [17]. We can in fact play • The regime with κa κn (first column with with two parameters, the ratio κn/κa and the average κa = 5 κn). In this case, we see that coherent states energy of the state hni. remain largely unaffected by the environment and 10

B. Approximate pointer states 0 0.1 0.2 0.3 0.4 For a general dynamics, exact pointer states, deﬁned κn/κa = 1/5 κn/κa = 1 κn/κa = 5 as states that do not get entangled with the environ- 5 ment if they are initially prepared, do not exist. Instead, t t t = = = we have to rely on an approximate notion of pointer τ 0 τ τ s s s states to give a meaningful notion of an emergent classi- −5 cal description. Approximate pointer states are deﬁned as the states that entangle the least with the environ-

t t ment according to a given entanglement measure. This t = =

5 =

τ τ definition of approximate pointer states, called the pre- ) c r τ c = α = dictability sieve and used in different contexts [35, 36], is = ( 0 τ τ r = c τ one definition among different proposal to formalize the /5 /5 −5 r idea of robustness against the environment. For instance, the Hilbert-Schmidt robustness criterion [37, 38] defines t −5 0 5 t = 5 = pointer states to be time-dependent pure states that best τ τ r c approximate according to the Hilbert-Schmidt norm the 5 = 0 5 = impure state generated by infinitesimal time evaluation. τ τ r

c Fortunately, those different approaches were shown to be −5 consistent for physically meaningful models [39]. −5 0 5 −5 0 5 Many different measures of entanglement exist in quan- <(α) tum information theory and it is not yet totally clear √ which one is the proper one to use in the predictability FIG. 6: Coherent state dynamics for α = 40 κn/κa. De- sieve approach. For the sake of simplicity, we will con- 2 pending on the relative ratio of the coupling constants κn/κa sider the purity γ defined as : we either see a emergent classical picture based solely on co- 2 herent states κn/κa < 1 or both coherent and then Fock states γ = tr ρ . (33) (κn/κa > 1). Approximate pointer states are then defined by searching for pure states that minimize the initial variation of the entropy or, in our case, maximize the variation of purity. evolves according to the dynamics of the L channel a It’s worth noting that the purity variation, for a pure with the characteristic dissipation timescale τ . r state, is also equal to twice the variation of the principal eigenvalue of the density matrix. • The complementary regime with κn κa (third Before proceeding, we could inquire about the depen- column with κn = 5 κa) where the Ln rules the dy- dence of the approximate pointer states that we find on namical evolution. We see that the coherent state the entanglement witness that we choose. For instance, is spread into a statistical mixture of Fock states we could have chosen a whole class of entropies Sα called 1 α over a timescale τc which is then followed by the the Rényi entropies [40] defined as Sα(ρ) = 1−α tr ρ . much slower process of relaxation. Using those entropies to find approximate pointer state do not change the conclusion if we are looking at the short-time evolution of a pure state. Indeed, if we pre- The intermediate regime with κ ≈ κ . In this • a n pare at t = 0 a pure state, we have that ρα = ρ for case, it is not possible to clearly conclude which α 6= 0. Then S˙ = α trρρ ˙ α−1 = α trρρ ˙ which is basis is the most classical one. α 1−α 1−α equal, apart from a proportionality factor, to the evolution of purity. The approximate pointer states do not The previous discussion focused mainly on a prepara- then depend on which measure we choose. tion of coherent states. A similar discussion can be made Coming back to the purity, its derivativeγ ˙ , which is, if we prepare a Fock state. As expected, if the Ln chan- up to a constant factor, just the derivative of the largest nel dominates, we observe a decoherence over the Fock eingenvalue for an initial pure state, is linear in ∂tρ. We basis with a slow relaxation towards the vacuum induce thus haveγ ˙ =γ ˙ a +γ ˙ n, whereγ ˙ a (resp.γ ˙ n) is the con- by the La channel. Nonetheless, some subtleties, detailed tribution solely due to the La (resp. Ln) channel. First in AppendixA2, occur in the opposite situation because of all, since each of these contributions is non positive, coherent states do not form a proper orthogonal basis. In a pure state will stay pure if and only if it stays pure the same way that a very close superposition of coherent states will not properly decohere under the influence of the La channel, a Fock state, being a continuous superposition of coherent states, cannot properly evolve toward 2 The purity can be used to define a notion of entropy S = 1 − γ a classical mixture of coherent states. called the linear entropy. 11

function of (κa, κn), can be seen for diﬀerent number of −0.2 0 0.2 0.4 0.6 photons on Fig.9. The time evolution of the approximate pointer states κa = 0 κa = 1 κa = 2 κa+κn κa+κn 5 κa+κn 5 is featured on Fig.8. This illustrates that on short 5 timescales, the optimal state keeps its shape and, as such, is quite robust to both interactions. 0 Furthermore, a remarkable fact can be seen by looking at the overlap between the Fock state of energy n0 and the optimal pure state |ψopti: Fock states remain the −5

) approximate pointer states even in the presence of the La

α κ 3 κ 4 κ ( a = a = a = 1 decoherence channel at small coupling κa κn. This is κa+κn 5 κa+κn 5 κa+κn = characterized by the presence of a plateau on the overlap 5 |hn0|ψopti| as a function of κa/(κa + κn) as shown on Fig.9. 0 To better understand this phenomenon, it is instruc- tive to look at the evolution of the purity for a state not −5 far away from the Fock state |n0i and see how this small perturbation changes or not the optimal problem. As- −5 0 5 −5 0 5 −5 0 5 P <(α) suming that the state |ψi = n cn |ni has coeﬃcients cn = δn,n0 + n, with |n| 1, we have:

X 2 2 FIG. 7: Wavepacket that minimizes the loss of purity at ini- γ˙ n = −2 κn |n| (n − n0) , (35) tial time, for an average energy of 20 photons, obtained by nu- n merical optimization techniques. The optimal (with respect 2 X 2 to the purity) stateγ ˙ opt interpolates between Fock states and γ˙ a = −2 κa n0(1 + 2<n0 ) + |n| (n + n0) coherent states depending on the relative value κa/(κa + κn) n of the coupling constants. 2 2 − |n0+1| (n0 + 1) − |n0−1| n0 − 2< ∗ pn (n + 1) . (36) n0+1 n0−1 0 0 for each channel. In this case, it is easy to see that, un- less one of the two constants κa or κn is zero, the only The norm and energy constraints can be rewritten as the state staying pure is the vacuum state. That’s why, in equation system: general, no exact pointer states exist for a complex open X 2 quantum dynamics and we have to look for approximate 2<n0 + |n| = 0 , (37a) ones. Having said that, we now have to search for pure n X 2 states that maximize the evolution of the purity at initial (n − n0)|n| = 0 . (37b) time. As a function of the matrix elements ρmn of the n initial state written in the Fock basis, the purities satisfy the equations: Using the global phase symmetry of the quantum state, we can fix n0 (an imaginary part on n0 corresponds to X γ˙ = −κ |ρ |2(m − n)2 , (34a) changing the phase of the initial Fock state). Further- n n m,n more, using both constraints, we can eliminate all the m,n terms containing n0 . The variation of purity takes the X h 2 γ˙ a = −κa |ρm,n| (m + n) form: m,n γ˙ X 2 2 ∗ p i = κan0 − |n| κn(n − n0) − 2ρm,nρm+1,n+1 (m + 1)(n + 1) . (34b) 2 n 2 2 + κa |n0+1| (n0 + 1) + |n0−1| n0 To find the approximate pointer states, we thus have to + 2< ∗ pn (n + 1) . (38) optimize the initial pure state that minimize the absolute n0+1 n0−1 0 0 purity variation, under the constraint of a unit norm. It is also natural to fix the average energy of the wavepacket We see that under the normalization constraint, Fock (otherwise, the vacuum is a trivial optimum). Since it states are always stationary points of the purity varia- is difficult to perform this optimization analytically, we tion. Furthermore, provided that κn > 4κa, the purity performed it numerically using the Pagmo library [41] variation is a maximum along all directions except possi- with the SNOPT algorithm [42]. We see on Fig.7 that, bly on the plane (n0−1, n0+1). We can thus concentrate when the average energy is a multiple of the number of on this plane. In this case, the constraint Eq. (37b) is 2 2 photons, the optimal state deforms from a coherent state simply |n0−1| = |n0+1| . If we denote θ the phase dif- to a Fock state. The optimal purity variation,γ ˙ opt as a ference between n0−1 and n0+1, the tipping point occurs 12

−0.2 0 0.2 0.4 0.6

(κa + κn)t = 0 (κa + κn)t = 1/10 (κa + κn)t = 1/2 (κa + κn)t = 2 (κa + κn)t = 10 5 κ a κ + a κ n

0 = 10 1

−5

5 κ a κ + a ) κ n α

( 0 = = 2 1

−5

5 κ a κ + a κ 0 n = 5 4

−5 −5 0 5 −5 0 5 −5 0 5 −5 0 5 −5 0 5 <(α)

FIG. 8: Evolution of the approximate pointer states for several (κa, κn) parameter regimes, for an initial energy of n0 = 20 photons.

thus: n0 1 2 5 10 1 1 κ /κ = n + + pn (n + 1) ≈ 2 n + . (40) n a 0 2 0 0 0 2 γ˙ opt/(κa + κn) |hn0|ψopti| 0 1 This equation gives the critical point where Fock states are no longer the einselected states of our dynamics and 0.75 explains the qualitative features of Fig.9. Actually, what −2 we just shown is that a Fock state is always a stationary 0.5 point of the purity variation. Given the norm and energy constraints, it moves from an extremum to a saddle p −4 point exactly when κn/κa = n0 + n0(n0 + 1) + 1/2, . 0 25 with n0 the average energy of the state. When n0 1, κn/κa ≈ 2n0 + 1. As such, when n0 becomes bigger, 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 the size of the plateau on which the Fock state remains κa/(κa + κn) the einselected state becomes smaller. Away from the critical point, the most robust state becomes the state that interpolate between a number and coherent state of FIG. 9: Purity and scalar product of the approximate pointer Fig.7. state with Fock states for energies of n0 = 1, 2, 5, 10 photons. We see that in the low κa regime, the approximate However, when the average number of photonn ¯0 in pointer state |ψopti is still a Fock state, until the critical value the cavity is not an integer, the behavior is quite dif- p κn/κa = n0 + n0(n0 + 1) + 1/2. As expected, the variation ferent. First, even when κa = 0, there is no exact of purity linearly decreases on this plateau. pointer state satisfying this constraint. The approximate pointer states in this case are superpositions of Fock states |ψsfi = α |bn0ci+β |dn0ei with (α, β) such that the when: energy constraint is satisﬁed. Another diﬀerence is that in this case, the optimal state deforms smoothly from the 1 p κ /κ = n + + n (n + 1) cos θ . (39) superposition |ψ i to the coherent state. Contrarily to n a 0 2 0 0 sf the integer case, there is no tipping point, as can be seen Since κn has to be maximum, we can keep θ = 0. And on Fig. 10. 13

remain robust over time (validity of the short-time hy- n¯0 1.5 2.5 5.5 10.5 pothesis)? As we already discussed, focusing on the short-time evolution basically solves the issue of which entanglement measure to choose, but the problem re- γ˙ opt/(κa + κn) |hψcoh|ψopti| |hψfs|ψopti| mains open in general. 0 1 1 From the exact solution of the model, we can compare the evolution over time of the purity between the opti- 0.8 0.8 mal, the Fock and the coherent state of a given energy −2 as shown in Fig. 11. As expected, we observe that, at short times, the evolution of purity is the slowest for the 0.6 0.6 optimal state (by construction) while, at very long time −4 compared to the relaxation time, everything converges 0.4 0.4 toward the same value since we are basically in the vacuum. However, at intermediate timescales, the evolution of purity gets quite involved and we see that it may even 0 0.5 1 0 0.5 1 0 0.5 1 be possible that the optimal state does not remain so and κa/(κa + κn) this is strongly dependent on the coupling constants. In this case, one can get an intuition of this behavior be- FIG. 10: Purity variation, and scalar product of the approxi- cause the optimal states have a smaller spreading on the mate pointer state with the approximate pointer states when Fock basis than the coherent states and as such, con- one of the coupling constant is zero. The average energy cor- tract to the vacuum slower. Thus, while the definition responds ton ¯0 = 1.5, 2.5, 5.5, 10.5 photons. In this case,√ the of approximate pointer states is physically intuitive, the optimal state when κa = 0 is |ψsfi = (|bn¯0ci + |dn¯0ei)/ 2. question remains on how to properly characterize them Conversely, when κn = 0, we have a coherent state whose from an information perspective but also from a dynam- average energy isn ¯0. Contrarily to the integer case, there is ical perspective. no plateau appearing in the low κa regime. A general question that we can also ask in the perspective we adopted here on understanding the emergence of a classical picture from a complex environment would be state |ψopti |αi |n0i the following: what can we learn about the dynamics and κa = 1 κa = 1 κa = 9 the einselection process of the model through the general κa+κn 10 κa+κn 2 κa+κn 10 1 features of the model like the algebraic relations between the jump operators? Naturally, the whole problem de- 0.75 pends on the kind of dynamical approximations we do, if we start from the exact Hamiltonian dynamics or from an γ 0.5 approximate master equation like the Lindblad equation 0.25 as we did and still have in mind here. Focusing on the general dynamics first, we saw that, 0 given the Lindblad equation and the commutation rela- . . 0 5 10 15 0 1 2 3 4 0 0 5 1 1 5 2 tions of the jump operators, the dynamics can be exactly (κa + κn)t solved algebraically by adopting a quantum channel perspective: the full dynamics of the model decouples and FIG. 11: Evolution of the purity for the coherent state, the we can look at the evolution of both quantum channels Fock state and the approximate pointer state for the energy separately, which is an easy task. In fact, our solution can n0 = 20 photons. be abstracted in the following sense. Consider an open quantum dynamics with a damping channel encoded by the jump operator L and a number decoherence channel V. DISCUSSIONS AND CONCLUSION encoded by the jump operator L†L. Our approach can then be followed steps by steps again. It however opens We discussed here the einselection process in the pres- the question on how to defined a proper phase space asso- ence of two incompatible decoherence channels and used ciated to a jump operator (through, for instance, gener- the characterization of approximate pointer states as alized coherent states [43]), how to generalize character- states that entangle the least with the environment. By istic functions like the Wigner representation and verify choosing an entanglement measure, they are found by if the phase space perspective on decoherence developed solving an optimization problem from the short-time throughout our analysis still holds in this generalized con- open evolution of pure states (under natural constraints). text. Two drawbacks of this approach to the einselection prob- Concerning the einselection process however, note that lem can be stated: how the choice of the entanglement having a general solution does not mean that finding the measure influences the answer and does an optimal state approximate pointer states is also solved. The problem 14 remains here largely open when we have many incom- Acknowledgments patible decoherence channels. Indeed, not only does the algebraic structure influences the einselection process but The authors thank Ekin Ozturk for his help with also the set of coupling constants and how they run as a Pagmo. We also thank Pascal Degiovanni for fruitful function of the energy [16, 17]. discussions and his comments on the manuscript.

We can start to grasp those subtleties already when Appendix A: Details on the Wigner representation a thermal environment is present. Indeed, absorption processes can occur which forces us to consider a jump √ † 1. Decoherence on the Fock basis operator of the form La† = κan¯ a . However, such a jump taken alone is hardly meaningful (the state evolves toward an infinite energy configuration). Relaxation pro- If decoherence occurs on the Fock basis, the compo- cesses have to be taken into account and this is summed nents are modulated by an exponential factor with a up by the usual commutation relations between a and decohrence timescale inversely proportional to the “dis- a†. Still, this is not sufficient to reach equilibrium and tance” between two Fock states. We can then write the proper relations between the coupling constants of the Wigner function as: different processes must exist. This well known exam- 2 X −κnl t/2 ilθ ple already shows that relating the general features of Wt(r, θ) = Wl(r)e e (A1) the open quantum dynamics to the einselection process l∈Z 2π 0 0 is not that straightforward. 0 θ − θ κnt dθ = W0(r, θ ) ϑ ; i (A2) ˆ0 2π 2π 2π +∞ 0 0 0 The model studied in this paper behaves quite intu- 2 = W0(r, θ ) Nσ =κnt (θ − θ ) dθ , (A3) itively: when one of the coupling dominates, the associ- ˆ−∞ ated decoherence channel controls the einselection pro- where W (r) is the lth harmonics of the angle vari- cess. However, when both couplings are comparable, the l able of the Wigner function at time t = 0, ϑ(z; τ) = non-trivial commutation relations between the jump op- P exp(πin2τ + 2πinz) is the Jacobi theta function erators enter the game. No exact pointer basis exists n∈Z and N 2 is the centered normal distribution of variance and the robust states interpolate between the extreme σ σ2. cases and how far they are from them depends on the In phase space, the dynamics induced by the L chan- relative values of the coupling constants again. Still, we n nel is thus the expected Gaussian spreading of a diffusive also unraveled the fact that, given some constraints, the evolution. When κ t 1, the periodicity can be forgot- transition from one class of pointer states to another as n ten. However, when κ t 1, we have the interesting a function of the relative values of the couplings is not n expansion of the Jacobi function: smooth: below a critical value, Fock states remains exactly the most robust states. How can such a behavior be n2 θ κnt X anticipated from the structure of the dynamics remains ϑ ; i = 1 + 2 e−κnt/2 cos nθ . (A4) 2π 2π ∗ to be explored. In the end, this shows that the question n∈N of predicting general features of the emergent classical picture only from the structure of the interaction (alge- Consequently, for κnt = 2, only the first harmonics dom- braic relations between jump operators, set of coupling inates with amplitude oscillations of 74 % of the average constants) still calls for a deeper understanding. value, as can be seen on Fig.4. Note that, if there is a discrete symmetry in the angle variable of the Wigner function, such as W (r, θ+2π/p) = In summary, we solved exactly a model of decoherence W (r, θ) for an integer p ≥ 2, the first non-zero modula- 2 for an open quantum system composed of two incom- tion term scales as e−κntp /2. Here, the scaling is not lin- patible decoherence channels using quantum trajectories ear but quadratic. This could be in fact directly recovered and phase-space techniques. We then studied numeri- by noting that , the symmetry W (r, θ + 2π/p) = W (r, θ) cally the dynamical emergence of a classical picture. We is equivalent to having ρk,k+l = 0 if l is not a multiple of were then able to see how the selection of approximate p. In the end, initial states possessing a symmetry will pointer states depends on the relative values of the cou- thus form a crown much quicker than the ones that don’t. pling constants. This unraveled the remarkable robustness of Fock states relative to a decoherence on the coherent state basis and we were able to analyze quantitatively 2. Decoherence on the coherent states basis the critical coupling where this robustness gets lost. Our results show that the physics of decoherence and einse- If we initially prepare the cavity in the Fock state |mi, lection still has a lot to offer when a complex dynamics then the Ln dynamics is trivial. As we have seen on is at play. Eq. (12), the state inside the cavity can be described as 15

this case, it corresponds to the time it takes for the prob- κ t = 0 κ t = 1/n κ t = 1/2 a a 0 a ability distribution to spread over several Fock states. In n = 8 n = 16 n = 32 the Wigner function, it corresponds to the attenuation of 0 0 0 the oscillations near the origin (see Fig. 12). 0.6 On the contrary, this does not correspond to a fast decoherence over coherent states. Since the Fock states

) 0.4 | have no deﬁnite phase, it is natural to look for a mixture α |

( 0.2 of coherent states which is uniform in its phase distribu- 0

n tion. A natural expression for any state without phase

W 0 preference would thus be:

−0.2 √ √ ρ = q(n ) | n eiθih n eiθ| dθdn . (A6) 0 1 2 3 0 1 2 3 4 0 2 4 6 ˆ 0 0 0 0 |α| We can easily show that FIG. 12: Radial part of the Wigner function for diﬀerent initial Fock states |n0i. X ρ = q(n )P (n) dn |nihn| , (A7) ˆ 0 n0 0 n a classical mixture of Fock state, the probability to have where P is the Poisson distribution of rate n . The the state |ki being given by: n0 0 resulting distribution over Fock states thus has a variance m equal to its mean. This is approximately the case for the p = (1 − e−κat)(m−k)e−κakt , (A5) k k binomial distribution only when p ' 0. As such, the timescale to have decoherence over coher- which is a binomial distribution of parameters m and ent states from an initial Fock states is 1/κa, the same as p = e−κat. The mean of such a distribution is mp and its the relaxation scale. This situation is actually similar as 2 variance is σ = mp(1 − p). the one for coherent states in respect to the Ln dynam- 2 For small times, we have σ ' mκat. This timescale ics: the short timescale governs the spreading in phase is comparable to the decoherence timescale τa we intro- space, while the long timescale governs the decoherence duced in Eq. (18) for the coherent states. We see that in between the Fock states. This is summarized on TableI.

[1] W. Zurek, Rev. Mod. Phys. 75 (2003). [15] J. M. Torres, R. Betzholz, and M. Bienert, Journal of [2] W. Zurek, arXiv preprint arXiv:0707.2832 (2008). Physics A: Mathematical and Theoretical 52, 08LT02 [3] W. Zurek, Nature Physics 5, 181-188 (2009). (2019), URL https://doi.org/10.1088%2F1751-8121% [4] M. Zwolak, H. Quan, and W. H. Zurek, arXiv preprint 2Faafffe. arXiv:0911.4307 (2009). [16] A. H. C. Neto, E. Novais, L. Borda, G. Zaránd, and I. Af- [5] C. J. Riedel and W. H. Zurek, Physical review letters fleck, Physical review letters 91, 096401 (2003). 105, 020404 (2010). [17] E. Novais, A. H. Castro Neto, L. Borda, I. Affleck, and [6] C. J. Riedel and W. H. Zurek, New Journal of Physics G. Zarand, Phys. Rev. B 72, 014417 (2005), URL https: 13, 073038 (2011). //link.aps.org/doi/10.1103/PhysRevB.72.014417. [7] C. J. Riedel, W. H. Zurek, and M. Zwolak, New Journal [18] V. Bonzom, H. Bouzidi, and P. Degiovanni, The Euro- of physics 14, 083010 (2012). pean Physical Journal D 47, 133 (2008). [8] C. J. Riedel, W. H. Zurek, and M. Zwolak, Physical Re- [19] A. N. Jordan and M. Buttiker,¨ Physical review letters view A 93, 032126 (2016). 95, 220401 (2005). [9] F. G. Brandão, M. Piani, and P. Horodecki, Nature com- [20] R. Ruskov, A. N. Korotkov, and K. Mølmer, Physical munications 6, 7908 (2015). review letters 105, 100506 (2010). [10] P. A. Knott, T. Tufarelli, M. Piani, and G. Adesso, Phys- [21] S. Hacohen-Gourgy, L. S. Martin, E. Flurin, V. V. Ra- ical review letters 121, 160401 (2018). masesh, K. B. Whaley, and I. Siddiqi, Nature 538, 491 [11] X. Qi and D. Ranard, arXiv preprint arXiv:2001.01507 (2016). (2020). [22] A. Essig, Q. Ficheux, T. Peronnin, N. Cottet, R. Les- [12] E. Arthurs and J. Kelly, The Bell System Technical Jour- canne, A. Sarlette, P. Rouchon, Z. Leghtas, and nal 44, 725 (1965). B. Huard, Multiplexed photon number measurement [13] H.-J. Briegel and B.-G. Englert, Phys. Rev. A (2020), 2001.03217. 47, 3311 (1993), URL https://link.aps.org/doi/10. [23] S. Haroche and J.-M. Raimond, Exploring the quantum: 1103/PhysRevA.47.3311. atoms, cavities, and photons (Oxford University Press, [14] J. M. Torres, Physical Review A 89, 052133 (2014). USA, 2013). 16

[24] G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, [35] C. Kiefer, I. Lohmar, D. Polarski, and A. A. Starobinsky, J. Raimond, and S. Haroche, Nature 400, 239 (1999). Classical and Quantum Gravity 24, 1699 (2007). [25] C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, [36] C. J. Riedel, Physical Review A 93, 012107 (2016). S. Kuhr, M. Brune, J.-M. Raimond, and S. Haroche, Na- [37] N. Gisin and M. Rigo, Journal of Physics A: Mathemat- ture 448, 889 (2007). ical and General 28, 7375 (1995). [26] A. O. Caldeira and A. J. Leggett, Physical Review A 31, [38] M. Busse and K. Hornberger, Journal of Physics A: 1059 (1985). Mathematical and Theoretical 43, 015303 (2009). [27] J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. [39] L. Diósi and C. Kiefer, Physical review letters 85, 3552 68, 580 (1992), URL https://link.aps.org/doi/10. (2000). 1103/PhysRevLett.68.580. [40] A. Rényi et al., in Proceedings of the Fourth Berkeley [28] Q. Ficheux, S. Jezouin, Z. Leghtas, and B. Huard, Nature Symposium on Mathematical Statistics and Probability, communications 9, 1926 (2018). Volume 1: Contributions to the Theory of Statistics (The [29] T. F. Havel, Journal of Mathematical Physics 44, 534 Regents of the University of California, 1961). (2003). [41] F. Biscani, D. Izzo, W. Jakob, GiacomoAcciarini, [30] H. J. Carmichael, Statistical Methods in Quantum Optics M. Märtens, M. C, A. Mereta, C. Kaldemeyer, S. Lyskov, 1: Master Equations and Fokker-Planck Equations, vol. 1 G. Acciarini, et al., esa/pagmo2: pagmo 2.12.0 (2019), (Springer Science & Business Media, 1999). URL https://doi.org/10.5281/zenodo.3582877. [31] V. I. Arnol’d, Mathematical methods of classical mechan- [42] P. E. Gill, W. Murray, and M. A. Saunders, SIAM review ics, vol. 60 (Springer Science & Business Media, 2013). 47, 99 (2005). [32] E. Wigner, Physical Review 40, 749 (1932). [43] A. Perelomov, Generalized coherent states and their ap- [33] J. Ville, Câbles et transmissions 2, 61 (1948). plications (Springer Science & Business Media, 2012). [34] P. Flandrin, Time-frequency/time-scale analysis, vol. 10 (Academic press, 1998).