The theory of quantum jump trajectories provides a new framework for understanding dynamical phase transitions in open systems. A candidate for such transitions is the atom maser, which for certain parameters exhibits strong intermittency in the atom detection counts and has a bistable stationary state. Although previous numerical results suggested that the “free energy” may not be a smooth function, we show that the atom detection counts satisfy a large deviations principle and, therefore, we deal with a phase crossover rather than a genuine phase transition. We argue, however, that the latter occurs in the limit of an infinite pumping rate. As a corollary, we obtain the central limit theorem for the counting process. The proof relies on the analysis of a certain deformed generator whose spectral bound is the limiting cumulant generating function. The latter is shown to be smooth so that a large deviations principle holds by the Gärtner–Ellis theorem. One of the main ingredients is the Krein–Rutman theory, which extends the Perron–Frobenius theorem to a general class of positive compact semigroups.

The last couple of decades have witnessed a revolution in the experimental realization of quantum systems.1 Ultracold atomic gases are created and used for the study of complex many body phenomena, such as quantum phase transitions2 shedding light on open problems in condensed-matter physics.3 

Real quantum systems are “open” in the sense that they interact with their environment, which leads to an irreversible loss of coherence and to energy dissipation. In many cases, the dynamics can be well described by the Markov approximation in which the environment possesses no memory and interacts weakly with the system. The joint unitary evolution of the system and environment can be described through the input–output formalism4 using a quantum stochastic calculus.5 In this framework, the Markov semigroup can be seen as the result of averaging over stochastic quantum trajectories arising from continuous-time measurements performed in the environment. These are described via stochastic Schrödinger (or filtering) equations6,7 and capture the system’s evolution conditional on the detection record.

In Ref. 8, a new perspective was put forward, which looks at quantum jumps from the viewpoint of non-equilibrium statistical mechanics.9 Detection trajectories are seen as “configurations” of a stochastic system, and large deviations theory10,11 is employed to study the dynamical phase transitions arising in this way. Consider for simplicity the case of a counting measurement, which is directly relevant for the model studied in this paper. The interesting scenarios are that of a phase crossover in which the counting trajectories show intermittency between long active periods (many counts) and passive ones (few counts) and that of phase coexistence where the counting process exhibits a mixture of infinitely long trajectories of either type. In the latter case, the asymptotic cumulant generating function (or “free energy”) of the total counts process Λt is singular at the origin, and the total counts do not obey a large deviations principle (LDP). In contrast, in a phase crossover, an LDP may hold, but numerically and practically, there would be a strong resemblance to an actual phase transition.

For finite-dimensional systems, the counting process Λt satisfies an LDP when the Markov dynamics is mixing, i.e., irreducible and aperiodic.12 The proof uses the Gärtner–Ellis theorem according to which it suffices to prove the convergence of the cumulant generating function to a smooth limit. By the Markov property, the former can be expressed in terms of a certain “deformed generator” Ls, and the existence of the limit

limt1tlogEρ(esΛt)=limt1tlogTr(ρetLs(1))=λ(s)

follows from the spectral gap property of Ls, where λ(s) is the spectral bound of Ls and ρ is the initial state.

In this paper, we investigate the existence of dynamical phase transitions for the atom maser, a well-known quantum open system exhibiting interesting properties such as bistability and sub-Poissonian statistics.13–15 The maser consists of a beam of excited atoms passing through a cavity with which they interact according to the Jaynes–Cummings model. After the interaction, the atoms are measured in the standard basis and the trajectory of measurement outcomes is recorded. For certain values of the interaction strength, the stationary mean photon number changes abruptly (cf. Fig. 1), and the distribution is bistable, having a low and a high energy “phase.” The measurement trajectories alternate between periods of low and high ground state atom counts (cf. Fig. 4), and its limiting moment generating function exhibits characteristic phase separation lines (cf. Fig. 5).

FIG. 1.

Mean photon number (black line) and photon number distribution (background) in the stationary state ρss as a function of α=Nexϕ.

FIG. 1.

Mean photon number (black line) and photon number distribution (background) in the stationary state ρss as a function of α=Nexϕ.

Close modal

Our main result (Theorem 5) is that the count process satisfies the LDP, and therefore, the atom maser does not have the non-analytic properties characteristic of phase transitions, although it exhibits clear phase crossover(s), which become sharper with the increasing pumping rate. As a corollary, we obtain the central limit theorem for the counting process using a result of Ref. 16. The proof follows the line of Ref. 12, but the novelty here is the treatment of an infinite-dimensional system in continuous time dynamics. We use an L2-representation17,18 of the semigroup generated by Ls and show that the corresponding semigroup is compact. We, then, use the Krein–Rutman theory (Ref. 19 and references therein) to establish the uniqueness and strict positivity of the eigenvector of λ(s) and, hence, the existence of the spectral gap. Some steps of the proof rely on a special feature of the maser dynamics, which allows us to restrict the attention to the commutative invariant algebra of diagonal operators. However, the line of the proof is applicable to general infinite-dimensional quantum Markov dynamics.

For recent work on quantum dynamical phase transitions, we refer to Refs. 8 and 20–23. In particular, our investigation was motivated by the numerical results of Ref. 24, indicating a possible non-analytic behavior of λ(s). In Ref. 12 (see also Ref. 25), a large deviations principle is shown to hold for correlated states on quantum spin chains; large deviations for quantum Markov semigroups are studied in Ref. 26. Metastable behavior in a different atom maser was investigated in Ref. 27. More broadly, there is a large body of large deviations work in quantum systems.28–33 

In Sec. II, we introduce the background of our problem: the atom maser and its Markov semigroup, the counting processes associated with the jump terms in the Lindblad generator, the static and dynamical phase transitions and the interplay between them, and the general setup of large deviations theory. In addition, the existence and properties of various semigroups are established rigorously. In Sec. III, we formulate the large deviations results and give a point-by-point outline of the proof. The results of a detailed numerical analysis are presented in Sec. IV, where we argue that “phase transitions” do occur in the limit of a very large pumping rate, at α ≈ 1 (second order), at α ≈ 6.66, and further points (first order), where α is the pumping parameter (see Fig. 1).

In this section, we introduce the atom maser dynamics, investigate the counting process associated with the measurement of outgoing atoms, and describe the basic elements of large deviations theory used in this paper. Propositions 2 and 3 establish the mathematical properties of the quantum dynamical semigroups used in this paper.

In the atom maser, two-level atoms pass successively through a cavity and interact resonantly with the electromagnetic field inside the cavity. The two-level atoms are identically and independently prepared in the excited state, and for simplicity, we assume that only a single atom passes through the cavity at any time. In addition, the cavity is also coupled to a thermal bath, which represents the interaction between the (non-ideal) cavity and the environment. The combined effects of the interactions with the atoms and the environment change the state of the cavity, which is described by a quantum Markov semigroup in a certain coarse-grained approximation described below (see Refs. 18 and 17 for a mathematical overview and Ref. 34 for the physical derivation of the master equation). In this section, we give an intuitive description of the dynamics starting with a simplified discrete time model, with an emphasis on the statistics of measurements performed on the atoms.

The cavity is described by a one mode continuous variable system with Hilbert space h=2(N) whose canonical basis vectors (|en)n0 represent pure states of a fixed number of photons. Therefore, if |ψh is a pure state, the photon number distribution of the cavity is given by |en,ψ|2. Mixed states are described by density operators, i.e., trace-class operators ρL1(h), which are positive and normalized to have unit trace, and the observables are represented by self-adjoint elements of the von Neumann algebra of bounded operators B(h) whose predual is L1(h).

Recall that the annihilation operatora on h is defined by

a|en=n|en1ifn>0,0ifn=0,

its adjoint is the creation operatora*, and N = a*a is the photon number operator such that N|en⟩ = n|en⟩. For every β > 0, we introduce the notation

D(Nβ)=u=n=0+un|en:n=0+n2β|un|2<+

for the domain of Nβ, and we recall that D(a)=D(a*)=D(N12). The atom is modeled by a two-dimensional Hilbert space C2 with standard orthonormal basis |0,|1 consisting of the “ground” and “excited” states. We denote by σ* and σ the corresponding raising and lowering operators (i.e., σ*|0⟩ = |1⟩). The interaction between an atom and the cavity is described by the Jaynes–Cummings Hamiltonian on C2h,

Hint=g(σa*+σ*a),

where g is the coupling constant. The free Hamiltonian is given by

Hfree=ω1a*a+ωσ*σ1,

where ω is the frequency of the resonant mode; however, by passing to the interaction picture, the effect of the free evolution can be ignored. Therefore, if the interaction lasts for a time t0, the joint evolution is described by the unitary operator U ≔ exp(−it0Hint) whose action on a product initial state is given by

U:|1|kcos(ϕk+1)|1|k+isin(ϕk+1)|0|k+1,

where ϕt0g is the accumulated Rabi angle. If a measurement is performed on the outgoing atom in the standard basis, then the cavity remains in state |k⟩ with probability cos2(ϕk+1) or gains an excitation with probability sin2(ϕk+1). By averaging over the outcomes, we obtain the cavity transfer operator T*:L1(h)L1(h),

T*(ρ)=K1ρK1*+K2ρK2*=K1*(ρ)+K2*(ρ),
(1)

where the Kraus operatorsKi are given by

K1=a*sin(ϕaa*)aa*,K2=cos(ϕaa*),

and Ki* are the corresponding jump operators on the level of density matrices. Since each atom interacts with the cavity only once, the state of the cavity after n such interactions is given by ρ(n)=T*n(ρ), which can be interpreted as a discrete time quantum Markov dynamics. Let us imagine that after the interaction, each atom is measured in the standard basis and found to be either in the excited or the ground state. The master dynamics can be unraveled according to these events as

T*n(ρ)=i=(i1,,in)Kin*Ki1*(ρ),
(2)

where each term of the sum represents the (unnormalized) state of the cavity after a certain sequence i = (i1, …, in) ∈ {0,1}n of measurement outcomes, whose probability is given by

Pρ(i1,,in)=Tr(Kin*Ki1*(ρ)).

If Λn(i) ≔ No. {j : ij = 0} denotes the number of ground state atoms detected up to time n, we can use the previous relation to compute its moment generating function,

EρesΛn=k0PρΛn=kesk=iesΛn(i)Tr(Kin*Ki1*(ρ))=Tr(T*sn(ρ)),
(3)

where

T*s(ρ)=esK1*(ρ)+K2*(ρ)

is a “deformed” transfer operator, i.e., a completely positive but not trace preserving map on L1(h). Relation (3) and its continuous time analog (14) will be the key to analyze the large deviations properties of the counting process in terms of spectral properties of operators, such as Ts and Ls.

To make the model more realistic, we will pass to a continuous time description in which the incoming atoms are Poisson distributed in time with intensity Nex, and the cavity is in contact with a thermal bath. If one ignores the details of short-term cavity evolution, the discrete time dynamics can be replaced by the coarse-grained continuous time Lindblad (master) equation,35 

ddtρ(t)=L*(ρ(t)),L*(ρ)=i=14LiρLi*12{Li*Li,ρ}=i=14LiρLi*+L*(0)(ρ)=i=14Ji*(ρ)+L*(0)(ρ),
(4)

with jump operators Li defined by

L1=Nexa*sin(ϕaa*)aa*,
(5)
L2=Nexcos(ϕaa*),
(6)
L3=ν+1a,
(7)
L4=νa*.
(8)

As before, the operators L1 and L2 are associated with the detection of an atom in the ground and excited state, respectively. The emission and absorption of photons due to the contact with the bath are represented by operators L3 and L4, respectively. Between jumps, the evolution is described by the semigroup etL*(0)(ρ)etG(ρ)etG with

G12i=14Li*Li=12Nex+ν+(2ν+1)N,D(G)=D(N).
(9)

Since we deal with an infinite-dimensional space and unbounded jump operators, the above definitions need to be formalized mathematically in order to ensure the existence and uniqueness of the different semigroups (see Proposition 2). As it is customary in the theory of quantum dynamical semigroups with an unbounded generator,36 so far, the generator L* can be safely defined on the linear manifold generated by the operators |u⟩⟨v| for u, vD(G) (this manifold is also a core due to Proposition 2 and Proposition 3.32 of Ref. 36), or equivalently, we can interpret L applied to any XB(h) as the sesquilinear form on D(G) × D(G) given by

u,L(X)v=Gu,Xv+u,XGv+i=04Liu,XLivu,vD(G).

Definition 1

(Ref. 36 and Sec. 3.1.2 in Ref. 37). Let B(h) be the space of bounded operators on h endowed with the w*-topology. A quantum dynamical semigroup on B(h) is a family S=(S(t))t0 of bounded operators on B(h) with the following properties:

  • S(0)=I.

  • S(s+t)=S(s)S(t) for all s, t ⩾ 0.

  • S(t) is completely positive for all t ⩾ 0.

  • S(t) is a w*-continuous operator on B(h) for all t ⩾ 0.

  • For each XB(h), the map tS(t)(X) is continuous with respect to the w*-topology on B(h).

The dynamical semigroup S(t) is called Markov (sub-Markov) if S(t)(1)=1(S(t)(1)1) holds true for every time t.

We recall that since the maps S(t) are positive, the fact that they are w*-continuous is equivalent for them to be normal [(Ref. 37, Lemma 2.4.19 and Theorem 2.4.21)]. The w*-generator Z is the operator defined as

Z(X)w*limh01hS(h)(X)X

for all XD(Z)XB(h):w*limh01hS(h)(X)X, which is a w*-dense linear space of B(h). Although no simple expression exists for the operators S(t) in terms of the generator Z, it is helpful to think of S(t) as the exponential of the generator

S(t)(X)=etZ(X),
(10)

especially from the point of view of relating spectral properties of Z to those of S(t), e.g., spectral mapping theorems. In Proposition 2, we show that the Heisenberg picture Lindbladian L is the generator of a quantum Markov semigroup (T(t))t0 on B(h); we postpone the Proof of Proposition 2 to  Appendix A.

Proposition 2.

  1. L generates a unique quantum Markov semigroup T(t), which has the following integral representation: for every XB(h),
    T(t)(X)=etL(0)(X)+k1i1,..ik=140t1tktX(t;t1,i1,,tk,ik)dt1dtk
    (11)
    and
    X(t;t1,i1,,tk,ik)e(ttk)L(0)Jike(t2t1)L(0)Ji1et1L(0)(X),
    where the equality is understood in terms of the associated bilinear form u,T(t)(X)v for u,vh.
  2. T(t)t0 has a unique faithful stationary state
    ρssρss(0)n0k=1nνν+1+Nexν+1sin2(ϕk)k|enen|
    (12)
    with ρss(0) taken such that Tr(ρss) = 1.
  3. T(t)t0 is ergodic, in the sense that any initial state ρ converges to the stationary state
    wlimtT*(t)(ρ)=ρss.

The dependence of the stationary mean photon number and photon number distribution on the “pumping parameter” αNexϕ is shown in Fig. 1 for ν = 0.15 and Nex = 150. We note two interesting features in Fig. 1: first, there is a sharp change in the mean photon number at α ≈ 1 followed by less pronounced jumps near α = 6.66 and α = 12. The other, related, feature to note is that the photon number distribution has a single peak for most values of α except in certain regions such as around the critical point α ≈ 6.66, where the stationary state has two local maxima. We will come back to these aspects in Sec. II B and show that they are related to features of the counting trajectories, such as intermittency, which indicates proximity to a dynamical phase transition. The reason for plotting the stationary distribution in terms of α (with Nex fixed) rather that ϕ is because the transitions appear to occur at fixed values of α and sharpen as Nex. This will be further investigated in Sec. IV.

To better understand the behavior of the stationary state illustrated in Fig. 1, we unravel the Markov semigroup T*(t) with respect to the four counting processes associated with the jump terms (5–8), each of them corresponding to a counting measurement of the quantum output process. If ρ is the initial state of the cavity, then ρ(t)T*(t)(ρ) is the evolved state at time t, which [in analogy to Eq. (2)] can be seen as an average over all possible counting events in the environment,

ρ(t)T*(t)(ρ)=etL*(0)(ρ)+k1i1,ik=140t1tktρ(t;t1,i1,tk,ik)dt1dtk,
(13)

where the integrand

ρ(t;t1,i1,,tk,ik)e(ttk)L*(0)Jik*e(t2t1)L*(0)Ji1*et1L*(0)(ρ)

is the unnormalized state of the cavity, given that detections of type i1, …, ik ∈ {1, 2, 3, 4} have occurred at times 0 ⩽ t1 ⩽ ⋯ ⩽ tkt, and no other counting events happened in the meantime. Its trace is interpreted as the probability of observing the given measurement record. Note that Eq. (13) can be obtained by duality from Eq. (11), so this description is mathematically rigorous. Among the four counting processes, we focus on the first one associated with the detection of an atom in the ground state and simultaneous absorption of a photon by the cavity. We denote by Λt the total number of such atoms detected up to time t: for every n1N,

Pρ(Λt=n1)=n2,n3,n40(***)0t1tkttr(ρ(t;t1,i1,,tk,ik))dt1dtk,

where (***) stands for

{i1,..in1++n4=1,,4:#{k:ik=j}=njj=1,,4}.

Similarly to the discrete case, by using the above unraveling and point 3 in Proposition 3, we can show that the moment generating function of Λt is given by

EρesΛt=TrT*s(t)(ρ)=TrρTs(t)(1),
(14)

where Ts(t)t0 is the quantum dynamical semigroup on B(h) with the generator

Ls(X)=esJ1(X)+i=24Jj(X)+L(0)(X)=(es1)J1(X)+L(X)
(15)

and T*s(t)t0 is the predual semigroup on L1(h). This is formalized in the following proposition whose proof can be found in  Appendix A.

Proposition 3.

For all sR, Ls generates a semigroup Ts=(Ts(t))t0 such that the following holds:

  1. Ts(t) is a quantum dynamical semigroup.

  2. Ts(t) is the unique solution to
    u,Ts(t)(X)v=etGu,XetGv+i=140tLierGu,Ts(tr)(X)LierGvdr
    (16)
    for every u,vh, where L1=es/2L1 and Li=Li for i = 2, 3, 4.
  3. Ts(t) has the integral representation
    Ts(t)(X)=etL(0)(X)+k1i1,..ik=140t1tktXs(t;t1,i1,,tk,ik)dt1dtk
    (17)
    for every t ⩾ 0, XB(h), and
    Xs(t;t1,i1,,tk,ik)e(ttk)L(0)Jike(t2t1)L(0)Ji1et1L(0)(X),
    where the equality has to be read for the associated bilinear form u,Ts(t)(X)v for u,vh and J1=esJ1, Ji=Ji for i = 2, 3, 4.

Equation (14) plays a central role in this paper; we will use it to formulate a large deviations principle for the counting process Λt and, in particular, to relate the moment generating function of Λt to the spectral properties of Ls. Note that Ls differs from the Lindblad generator by the factor es multiplying the jump term associated with the detection of a ground state atom. It is still the generator of a completely positive semigroup, but it is no longer identity preserving and, therefore, does not represent a physical evolution except for s = 0.

Unraveling (13) allows for a classical interpretation of the cavity dynamics. Indeed, the semigroup generated by L (and Ls) leaves invariant the commutative subalgebra Bd(h)B(h) generated by the number operator N, and the restriction of T(t)t0 to the diagonal algebra is the dynamical semigroup of a classical birth–death process on the state space {0, 1, 2, …}, with rates

λk2Nexsin(ϕk+1)2+ν(k+1),k0,μk2(ν+1)k,k1.
(18)

Figure 2 shows the birth and death rates (minus the common factor νk) rescaled by a factor Nex, in the limit Nex, as functions of the parameter ϑ(k+1)/Nexα=ϕk+1. In this regime, the rates become the functions λϑ2=sin(θ)2 and μϑ2=α2ϑ2 of the continuous parameterϑ, and we plot λϑ2 along with μϑ2 for different values of α. The intersection points correspond to minima and maxima of the stationary distribution35 as suggested by the following argument. For α < 1, the death rate is always larger than the birth rate and the distribution is maximum at the vacuum state. For 1 < α < 4.6, there is a single non-trivial intersection point such that the birth rate is larger to its left and smaller to its right and, therefore, corresponds to the maximum of the stationary distribution. Similarly, when 4.6 < α < 7.8, the rates intersect in three points: the first and last are located at local maxima while the middle point is a local minimum, so we deal with a bimodal distribution. However, while this analysis clarifies the emergence of multimodal distributions, it does not explain the sudden jump of the mean photon number at α ≈ 6.66 and higher values.

FIG. 2.

The birth (blue) and death rates as functions of ϑ for different values of α. The intersection points correspond to minima and maxima of the stationary distribution.

FIG. 2.

The birth (blue) and death rates as functions of ϑ for different values of α. The intersection points correspond to minima and maxima of the stationary distribution.

Close modal

This feature can be intuitively understood by appealing to the effective potential model.1 If we think of the photon number as a continuous variable and introduce a fictitious potential U defined by

ρss(n)=ρss(0)eU(n),
(19)

then the photon number distribution appears as the thermal equilibrium distribution of a particle moving in the potential U (with kB · T = 1); see Fig. 3. When the potential has a single local minimum (for 0 < α < 4.6), the stationary distribution is unimodal and concentrates around this point. The cavity state fluctuates around the mean, and Λt increases steadily with an average rate. When there are two (or more) local minima of different heights, the higher minimum corresponds to a metastable phase from which the system eventually escapes due to thermal fluctuations. The rate of return to the metastable phase is typically much lower due to the larger potential barrier that needs to be climbed. The point α ≈ 6.66 where the two local minima are equal plays the role of a “phase transition” and corresponds roughly to the point where the mean photon number changes abruptly. Here, the cavity spends long periods of time around the two local maxima with rare but quick transitions between them. The change from the low energy to the high energy mode is accompanied by a clear change in the slope of the counting process Λt.

FIG. 3.

Rescaled potentials U(n)/Nex as a function of n/Nex for various finite Nex converge to a limit potential for Nex. For α < 1, the potential is minimum at zero; for 1 < α < 4.6, it has a unique minimum away from n = 0; and for 4.6 < α < 7.8, there are two local minima, which become equal at α ≈ 6.66.

FIG. 3.

Rescaled potentials U(n)/Nex as a function of n/Nex for various finite Nex converge to a limit potential for Nex. For α < 1, the potential is minimum at zero; for 1 < α < 4.6, it has a unique minimum away from n = 0; and for 4.6 < α < 7.8, there are two local minima, which become equal at α ≈ 6.66.

Close modal

In the stationary regime, the mean Eρss(Λt) grows linearly with time with rate Tr(ρssL1*L1). This expression can be obtained by differentiating the moment generating function (14) at s = 0,

Eρ(Λt)t=1tddsEρ(esΛt)s=0=1tddsTr(ρTs(t)(1))s=0=1tu=0tduTr(ρTs(u)J1Ts(tu)(1))s=0=1tu=0tduTr(T*(u)(ρ)L1*L1),

where we used the fact that dLsdss=0=J1; cf. (15). The rate is, then, obtained by taking t and using the fact that ρ converges to the stationary state.

Using the property of the birth–death process nρss(n)(λn2μn2)=0, we can further write the rate as

Eρss(Λt)t=Nexnρss(n)sin2(ϕn+1)=nnρss(n)ν.
(20)

Unlike the “first order transition” occurring at α = 6.66, a “second order transition” occurs at α ≈ 1. Here, the first derivative of the mean photon number has a jump in the limit of Nex. This and the scaling of the potential U with Nex will be discussed in Sec. IV.

Equation (20) shows that the statistics of the trajectories are, therefore, closely related to the dynamics of the cavity and, consequently, to its stationary state. The next step is to think of the time trajectories as “configurations” of the stochastic system draw from ideas in non-equilibrium statistical mechanics and large deviations theory to study their phases and phase transitions.

The main result of this paper is the existence of a large deviations principle for the counting process Λt introduced above. Such results have already been obtained in the context of discrete time quantum Markov chains with finite-dimensional systems,12 but the novelty here is that we consider a continuous time Markov process with an infinite-dimensional system. The physical motivation lies in the new approach to the study of phase transitions for open systems developed in Refs. 8 and 24. Here, the idea is to identify dynamical phase transitions of the open system by analyzing the statistics of jump trajectories in the long time (stationary) regime. The trajectories play an analogous role in the configurations of a statistical mechanics model at equilibrium. In this analogy, the parameter s of the moment generating function (14) can be seen as a “field” that biases the distribution of trajectories in the direction of active or passive trajectories by effectively changing the probability of a trajectory ω ≔ (i1, t1, …, in, tn) by a factor exp(sΛt(ω)). When α is such that the stationary distribution is unimodal, the trajectories’ distribution changes smoothly from passive ones for s < 0 to active ones for s > 0. However, near α ≈ 6.66 (corresponding to the jump in the mean photon number), there is a steep change in the counting rates around s = 0. The active trajectories are associated with periods when the cavity is in the higher excited phase, while the passive trajectories are connected to the lower phase. Since the cavity makes very rare transitions between the phases, any trajectory—when followed for long but finite periods of time—falls typically into one of the two distinct categories (see Fig. 4). Our goal is to investigate whether this distinction survives the infinite time limit, in which case we would deal with a dynamical phase transition characterized by the non-analyticity of a certain large deviations rate function. We will show that this is not the case, but rather we deal with a crossover behavior, that is, the count rate does not jump but has a very steep change around s = 0, which appears to become a jump only in the limit of an infinite pumping rate Nex (see Sec. V).

FIG. 4.

Sample trajectories for the birth–death process describing the cavity state jumping on the ladder of Fock states |k⟩⟨k| (top left and right) and total measurement counts Λt (bottom left and right) for α ≈ 1 (left) and α ≈ 6.66 (right) at Nex = 50. The corresponding stationary state distributions (center) showing large variance at α ≈ 1 (red) and bistability at α ≈ 6.66 (green).

FIG. 4.

Sample trajectories for the birth–death process describing the cavity state jumping on the ladder of Fock states |k⟩⟨k| (top left and right) and total measurement counts Λt (bottom left and right) for α ≈ 1 (left) and α ≈ 6.66 (right) at Nex = 50. The corresponding stationary state distributions (center) showing large variance at α ≈ 1 (red) and bistability at α ≈ 6.66 (green).

Close modal

We will now briefly review some basic notions of large deviations needed in this paper. We refer the reader to Ref. 10 for a complete treatment; see Ref. 11 for a comprehensive overview and Ref. 38 for an introduction to large deviations in the context of statistical mechanics. Large deviations is a framework for studying rare events, more precisely events whose probabilities decay exponentially for a sequence of probability distributions. A key result is the Gärtner–Ellis theorem, which relates the rate of the exponential decay to the limiting behavior of the moment generating functions associated with the random variables.

Informally, a sequence (μn)nN of probability distributions on Rd endowed with the Borel σ-field B satisfies a large deviations principle (LDP) if there exists a function I:Rd[0,] such that

μn(dx)enI(x)dx.

More rigorously, the function I is called a rate function if it is lower semicontinuous [that is, its level sets xRd:I(x)α are closed]; if, in addition, its level sets are compact, we call it a good rate function. The domain of I is the set of points in Rd for which I is finite. The limiting behavior of the probability measures μn is characterized in terms of asymptotic upper and lower bounds on the values that μn assigns to measurable subsets ΓB. The sequence of probability measures μn satisfies a large deviations principle with a rate function I (or shortly, it satisfies an LDP) if for all ΓB,

infxΓ0I(x)lim infn1nlogμn(Γ)lim supn1nlogμn(Γ)infxΓ̄I(x).
(21)

Our goal is to prove an LDP for the counting process Λt of the atom maser; we will do this not by showing that Λt satisfies the above definition directly but by applying the Gärtner–Ellis theorem, which gives sufficient conditions on the sequence of probability measures in order to satisfy an LDP.

Theorem 4
(Gärtner–Ellis theorem, Ref. 10, pp. 44–55). Let (Zn)nN be a sequence of random variables in Rd with laws μn. Suppose that the (limiting) logarithmic moment generating function
λ(s)=limn1nlogEens,Zn,sRd,
exists as an extended real number and is finite in a neighborhood of the origin, and let λ* denote the Fenchel–Legendre transform of λ, given by
λ*(x)=supλRds,xλ(s).
If λ is an essentially smooth, lower semicontinuous function (e.g., λ is differentiable on Rd), then (Zn)nN satisfies a LDP with good rate function λ*.

The discrete index in the Gärtner–Ellis theorem can be replaced by a continuous one with the obvious modifications in (21). By the Gärtner–Ellis theorem, Λt satisfies an LDP if the following limit exists and is a differentiable function:

λ(s)limt1tlogEρesΛt=limt1tlogTrρTs(t)(1).
(22)

We will show that this is, indeed, true and λ(s) is spectral bound (i.e., the eigenvalue with the largest real part) of a certain generator Ls(d), which is closely related to Ls. An essential ingredient is the Krein–Rutman theory, which generalizes the Perron–Frobenius theorem to compact positive semigroups and ensures that λ(s) is real and non-degenerate. In particular, our analysis shows that λ(s) is smooth and its derivatives at s = 0 are the limiting cumulants of Λt,

limt1tCk(Λt)=dkλ(s)dsks=0,k1,

with the first two being the mean and the variance. Moreover, the generator Ls(d) has a non-zero spectral gap; this spectral analysis is illustrated in Fig. 5.

FIG. 5.

Derivative λ′(s) (a) and the spectral gap g(s) of Ls (b) as functions of s and α=ϕ/Nex (after Fig. 3 in Ref. 24).

FIG. 5.

Derivative λ′(s) (a) and the spectral gap g(s) of Ls (b) as functions of s and α=ϕ/Nex (after Fig. 3 in Ref. 24).

Close modal

Our main results are the following large deviations and central limit theorems. For reader’s convenience, we outline the key steps of the proofs below.

Theorem 5.

Suppose that the initial state ρ is a finite rank operator with respect to the Fock basis or, more generally, that n⩾0|⟨en, ρen⟩|2ρss(n)−1 < +. Then, the counting process Λt satisfies the large deviations principle with a rate function equal to the Legendre transform of λ(s), where λ(s) is the limit in (22). The function λ(s) is smooth, and it is equal to the spectral bound of a certain semigroup generator Ls(d) defined below.

In particular, the atom maser does not exhibit dynamical phase transitions, but rather crossover transitions, which become sharper as Nex increases. This behavior will be investigated in more detail in Sec. IV.

Corollary 6.
The counting process Λt satisfies the central limit theorem,
1t(Λttm)DN(0,V),
where D denotes the convergence in distribution and m and V are the mean and variance,
m=dλ(s)dss=0=Eρss(Λt)t,V=d2λ(s)ds2s=0.

Proof of Theorem 5 and Corollary 6.

For clarity of the exposition, we break the proof into individual steps.

  • We introduce L2(ρss) as the completion of B(h) endowed with the norm ‖·‖2 induced by the following inner product:
    Y,X=Tr((ρss1/4Yρss1/4)*(ρss1/4Xρss1/4)).
    (23)
    We recall (see Ref. 18, especially Proposition 2.1) that L2(ρss) is isomorphic as an Hilbert space to the Schatten ideal L2(Tr){XB(h):Tr(X*X)<+} via the unique continuous extension of the correspondence
    i:B(h)L2(Tr),Xρss1/4Xρss1/4.
    Hence, with an abuse of notation, we will identify YL2(ρss) with the corresponding operator in L2(Tr). We recall that for every Xn,XB(h) and YL2(ρss), the following holds:
    1. X2 ⩽ ‖X (Ref. 18, Proposition 2.1) and

    2. if Xnw*X, then ⟨Y, Xn⟩ → ⟨ Y, X⟩: indeed,
      limn+Y,Xn=limn+Tr(Y*ρss1/4Xnρss1/4)=limn+Tr(ρss1/4Y*ρss1/4Xn)=Tr(ρss1/4Y*ρss1/4X)=Y,X.

Lemma 7.

For every sR, the following holds:

  1. There exists a unique strongly continuous semigroup (Ts(t))t0 of bounded linear maps on L2(ρss) such that
    Ts(t)(X)=Ts(t)(X),XB(h).
    Every XD(Ls) belongs to the domain of the generator Ls of (Ts(t))t0 and
    Ls(X)=Ls(X),XD(Ls)=D(L).
    As usual, we denote (T(t))t⩾0 and L as the semigroup and the corresponding generator in the case s = 0.
  2. The set M(h) of finite rank operators given by finite matrices with respect to the Fock basis forms a core for Ls.

  3. Ls = L + δs, with δs being a bounded perturbation.

Proof of Lemma 7.

  1. First, we show that ρss is sub-invariant for T (as defined in the proof of points 1. and 2. of Proposition 3). Since ρssD(L*) and ρss is invariant for T, this is equivalent to showing that
    L*(ρss)=(L*L*)(ρss)=(es1)J1*(ρss)1s>0Nex1+1νρss0.
    This inequality is trivial for s < 0, while for s > 0, we can write the explicit form of ρss and get
    J1*(ρss)=Nexn0sin2(ϕn+1)ρss(n)|n+1n+1|=Nexn0sin2(ϕn+1)ρss(n)ρss(n+1)ρss(n+1)|n+1n+1|Nex1+1νρss
    since
    ρss(n)ρss(n+1)=(ν+1)(n+1)ν(n+1)+Nexsin2(ϕn+1)1+1ν.
    Then, we apply Theorem 2.3 of Ref. 18 and we obtain that the semigroup T can be extended to a strongly continuous contraction semigroup on L2(ρss). In addition, here, the conclusion follows multiplying the semigroup by a suitable exponential factor as for points 1. and 2. of Proposition 3.
  2. and 3. The proof of the previous point shows that Nex1+1ν1J1*(ρss)ρss; hence, we can apply again Theorem 2.3 of Ref. 18 in order to show that Nex1+1ν1J1 extends to a bounded operator on L2(ρss) and so does LsL=(es1)J1; let us call δs such extension. Since on M(h) we have Ls = L + δs, it follows that if we prove that M(h)D(L) is a core for Ls, we have that D(Ls) = D(L) and Ls = L + δs. Note that both L and δs preserve M(h) and that M(h) is dense in B(h) in the w*-topology and, hence, in L2(ρss) in norm. Hence, we only need to show that M(h) is a set of analytic vectors for Ls and, then, apply Proposition 14 in  Appendix B.

Let us fix sR. Then, for n,mN, we have
Ls(|enem|)=αm,n|en1em1|+βm,n|enem|+γm,n|en+1em+1|,
(24)
where
αm,n=νnm+Nexessin(ϕn)sin(ϕm),βm,n=12(ν+1)(m+n)+ν(m+n+2)+Nex(cos(ϕn+1)cos(ϕm+1)1),γm,n=(ν+1)n+1m+1.
Note that |αm,n|+|βm,n|+|γm,n|3(ν+1+2(es1)Nex)=:B(n+m+1). Hence,
Lsk|enem|Bk(n+m+1)(n+m+3)(n+m+2k1)Bk2k(n+m+1)(n+m+2)(n+m+k)=Bk2kk!n+m+kn+m2n+m4kBkk!.
If T̄=1/(4B), for any t<T̄, k=0+tkLsk(|enem|)k! converges uniformly on compact intervals in the uniform norm, hence, with respect to ‖‖2. By the definition of M(h), we can deduce the same for every XM(h). Let us call Xtk=0+tkLsk(X)k! and note that since the series of the derivatives converges uniformly on compact intervals, it solves the following abstract Cauchy problem:
Xt=X+Ls0tXudu,tT̄.
Hence, by Proposition 6.4 of Ref. 39, Xt = Ts(t) (X) for every t<T̄.

  • (ii)
    The moment generating function of Λt for an initial state ρ [cf. Eq. (14)] can be expressed in terms of the semigroup acting on L2(ρss) as
    Eρ(esΛt)=Tr(ρTs(t)(1))=ρ̃,Ts(t)(1),
    where ρ̃ρss1/2ρρss1/2 is assumed to belong to L2(ρss) or, equivalently, n⩾0|⟨en, ρen⟩|2ρss(n)−1 < + [in this case, ρ̃ extends to a bounded linear functional on L2(ρss)]. This holds, for instance, not only if ρ has a finite number of photons but also if ρ = ρss.
  • (iii)
    As we already mentioned, the commutative von Neumann algebra Bd(h) of the operators that are diagonal in the Fock basis plays a fundamental role. We need to introduce some other related linear spaces,
    Md(h)M(h)Bd(h)=span{|enen|:nN}D(L),Ld2(ρss)Md(h)̄2.
    Note that Bd(h)(N) as von Neumann algebras and Ld2(ρss)2(N,ρss) as Banach spaces.

Proposition 8.

The following statements hold for every sR.

  1. The generator Ls and the corresponding semigroup (Ts(t))t0 preserve the algebra Bd(h). Consequently, Ls and Ts also preserve the subspace Ld2(ρss).

  2. The action of Ls on the diagonal is explicitly written as an operator Ls(d) acting on f=kfk|ekek|D(Ls(d))D(Ls)Bd(h) as Ls(d)(f)=g=kgk|ekek| with
    gk=k(ν+1)(fkfk1)+(es1)Nexsin2(ϕk+1)fk+1+(ν(k+1)+Nexsin2(ϕk+1))(fk+1fk).
    (25)

In particular, L0(d)=L(d) is the generator of a birth–death process with birth and death rates as in Eq. (18).

Proof of Proposition 8.

  1. Md(h)D(Ls)Bd(h) is w*-dense in Bd(h). Hence, it is sufficient to compute explicitly Ls(|ekek|) and observe that it belongs to
    span{|ejej|,j=k1,k,k+1}
    for all k. Then, Ls(Md(h))Md(h). The rest follows from the definitions.
  2. This point is a direct computation by using the definition of the generator Ls given by Eqs. (4) and (15).□

We denote by Ls(d) and Ts(d) the restrictions of Ls and Ts to Ld2(ρss) (as usual, we drop the index s in the case s = 0). Since 1Ld2(ρss), the moment generating function can be expressed as
Eρ(esΛt)=ρ̃(d),Ts(d)(t)(1),
with ρ̃(d) denoting the diagonal of ρ̃.
  • (iv)

    The semigroup (Ts(d)(t))t0 is immediately compact [i.e., Ts(d)(t) is compact for all t > 0]. In order to prove it, we first show it for T(d)(t) with the classical theory of birth–death processes and then use perturbation theory.

Lemma 9.

The following statements hold.

  1. L(d) is self-adjoint, and its essential spectrum is empty.

  2. T(d)(t)t0 is immediately compact.

  3. Ts(d)(t)t0 is immediately compact for all sR.

We recall that σ(L(d)) is the disjoint union of the discrete and the essential spectrum and the discrete spectrum for a self-adjoint operator is defined as those αC, which are isolated eigenvalues with finite multiplicity (see Ref. 40, Theorem VII.10).

Proof of Lemma 9.

  1. The proof of point 2. of Lemma 7 shows that Md(h) is a dense linear space of analytic vectors for L(d); hence, by the Nelson theorem (Ref. 41, Theorem X.39), in order to establish the self-adjointness of L(d), it is enough to check that it is symmetric on Md(h) [which is a simple computation using Eq. (24)].

    In order to show that the essential spectrum of L(d) is empty, it suffices to check that the following condition on birth and death rates holds (cf. Ref. 42, Theorem 1.2) and we already know that the process is non-explosive and admits a unique invariant state,
    limn+i=1n1μi2ρss(i)j=n+1+ρss(j)=0.
    Let us do the following computations:
    i=1n1μi2ρss(i)j=n+1+ρss(j)=i=1nj=n+1+1μi2ρss(j)ρss(i)=i=1nj=n+1+1μi2k=i+1jν+Nexsin2(ϕk)/kν+1.
    Note that for every 0 < ϵ < 1, there exists MN such that for every kM, Nexsin2(ϕk)ϵk; therefore, for n > M, we have
    i=1n1μi2ρss(i)j=n+1+ρss(j)i=1M1μi2ρss(i)j=n+1+ρss(j)=o(1)+i=M+1n1μi2j=n+1+ν+ϵν+1=:qji111qi=M+1nqni(ν+1)i+o(1)=11qi=0nM1qi(ν+1)(ni)+o(1).
    The result follows from the fact that q < 1 and the dominated convergence theorem.
  2. L(d) is a self-adjoint, unbounded operator with an empty essential spectrum such that L(d) ⩽ 0; therefore, its spectral resolution reads
    L(d)=n0αnPn,
    where Pn’s are finite-dimensional orthogonal projections and αn’s are distinct non-negative real numbers such that limn→+αn = + (they do not accumulate). The semigroup generated by L(d) can be expressed via the functional calculus as
    T(d)(t)=n0etαnPn,t0.
    By its spectral representation, we can conclude that T(d) is immediately compact.
  3. The restriction Ls(d) is a bounded perturbation of the generator L(d); hence, the semigroup Ts(d)(t)t0 is also immediately compact; cf. Ref. 43, Theorem III.1.16.□

Since (Ts(d)(t)) is an immediately compact semigroup, we have (Ref. 43, Corollary IV.3.12) a spectral mapping theorem of the form
etσ(Ls(d))=σ(Ts(d)(t))\0,t>0.
In particular, the spectral radius of Ts(d)(t) is given by
rs(t)r(Ts(d)(t))=etλ(s),
where λ(s) is the spectral bound of Ls(d), i.e., the real part of the eigenvalue with the largest real part.
  • (v)

    The semigroup (Ts(d)(t))t0 is strictly positive, that is, Ts(d)(t)(D)>0 for all D ⩾ 0 in Ld2(ρss) and t > 0.

Proof.

It is not difficult to see that every D ⩾ 0 in Ld2(ρss) is of the form D = kDk|ek⟩⟨ek| for some Dk2(N,ρss) and Dk ⩾ 0 for every k; hence, it is enough to show that for every kTs(d)(|ekek|)>0 for every t > 0. Note that (Pm,n(t)et(1es)1+1νNex1{s>0}en,Ts(d)(t)(|emem|)en)n,mN is a standard transition function, and so by the well-known results of continuous time Markov chains, we have that either Pm,n is constant and equal to 0 or it is strictly positive for every t > 0 [Levy’s theorem (Ref. 44, Proposition 1.3)]. Since the birth and death rates are all strictly positive, Pm,n(t) > 0 for every positive time (for a reference about continuous time Markov chains, see, for instance, Ref. 44).□

  • (vi)

    Since Ts(d)(t)t>0 is compact and strictly positive, Krein–Rutman theory (Ref. 19, Theorem 1.5) implies that the spectral radius of Ts(d)(t) is an algebraically simple eigenvalue with strictly positive right and left eigenvectors r(s) and l(s).

Lemma 10.
For every sR, there exists a strictly positive number g(s) > 0 such that for every t > 0,
Ts(d)(t)=etλ(s)(|r(s)l(s)|+Rts)
(26)
and RtsLd2(ρss)Ld2(ρss),=O(eg(s)t).

Proof of Lemma 10.
Because of the compactness of Ts(d)(t), we only need to prove that
σ(Ts(d)(t)){zC:|z|=eλ(s)t}={eλ(s)t}.
Suppose that there exist θ ∈ [0, 2π) and DLd2(ρss) such that Ls(d)(D)=(λ(s)+iθ)D; then, if we consider t = 2π/θ, Ts(d)(t)(D)=eλ(s)tD. Since l(s) is the unique eigenvector with eigenvalue one, we conclude that θ = 0.□
Using point (iii), this implies that
Eρ(esΛt)=etλ(s)ρ̃,r(s)l(s),1+o(1).
(27)
Since l(s), r(s) > 0 and ρ̃,10, the inner products are non-zero and we obtain the limiting cumulant generating function
limt1tlogEρin(esΛt)=λ(s).
  • (vii)
    Using analytic perturbation theory for the generator Ls(d), the spectral bound λ(s) can be shown to be a smooth function of s in a complex neighborhood of the real line. Since for every s0R, λ(s0) is an isolated eigenvalue with finite multiplicity, we can apply Proposition 3.25 of Ref. 45, p. 141 to the family of perturbations
    Vss0Ls(d)=Ls0(d)+δsδs0=Ls0(d)+es0(ess01)J1(d)
    and we find that λ(s) is an analytic function of s and remains isolated in a complex neighborhood of s0.
  • (viii)

    Using points (vi) and (vii), we apply the Gärtner–Ellis theorem to conclude that Λt satisfies the LD principle with the rate function equal to the Legendre transform of λ(s).

  • (ix)
    Furthermore, by the result of Ref. 16, it follows that Λt satisfies the CLT. In particular, the limiting cumulants of Λt can be computed as derivatives of λ(s) at s = 0,
    limt1tCk(Λt)=dkλ(s)dsks=0.
    By J1(d), we denote the operator acting on D=kDk|ekek|Ld2(ρss) in the following way:
    J1(d)(D)=k1Nexsin2(ϕk)Dk|ek1ek1|.

Lemma 11.
dλ(s)dss=0=1,J1(d)(1),d2λ(s)ds2s=0=1,J1(d)(1)+21,J1(d)(DV),
and DV can be characterized as the unique solution in Ld2(ρss) of
L(d)(DV)=1,J1(d)(1)1J1(d)(1).

Proof of Lemma 11.
Differentiating first once and then twice
Ls(d)(r(s))=λ(s)r(s)
and evaluating in s = 0, we get
L(d)(r(0))+J1(d)(1)=λ(0)1,
(28)
L(d)(r(0))+J1(d)(1)+2J1(d)(r(0))=λ(0)1+2λ(0)r(0),
(29)
where we used the fact that λ(0) = 0 and r(0) = 1. Note that l(0) = 1 too; hence, taking the scalar product of Eq. (28) against 1, we get
λ(0)=1,J1(d)(1).
Substituting the expression we obtained for λ′(0) in Eq. (28), we get that r′(0) is the unique [remember that ker(L(d)) has dimension 1] solution of
L(d)(r(0))=1,J1(d)(1)1J1(d)(1).
We can choose r(s) such that ⟨1, r(s)⟩ = 1; hence, substituting the expression we obtained for λ′(0) in Eq. (29) and taking the scalar product against 1, we get
λ(0)=1,J1(d)(1)+21,J1(d)(r(0)).
Note that λ′(0) is the expected value of Λtt if the system starts in the stationary state ρss [Eq. (20)]; indeed,
1,J1(d)(1)=Tr(ρssL1*L1)=n0nρss(n)ν.

The existence of a “phase transition” in the atom maser has been discussed in several theoretical physics papers.13,24,35,46,47 There is a general agreement that if Nex is sufficiently large (for instance, Nex ≈ 150), then “for all practical purposes,” we can consider that the mean photon number of the stationary state has a jump at α ≈ 6.66 (see Fig. 1), which matches up with a jump between the left and right derivatives of λ(s) at s = 0 in the dynamical scenario (see Fig. 5). However, the question whether we are dealing with a “true” (dynamical) phase transition or rather a steep but smooth crossover was left open and motivated this investigation. Having proved that the latter is the case, we would like to briefly put the result in the context of a numerical analysis.

As the proof suggests, dynamical phase transitions are intimately connected with the closing of the spectral gap of the semigroup generator. Figure 5 shows the close match between the behavior of the first derivative of λ(s) and the spectral gap g(s) ≔ λ(s) − Reλ1(s). In particular, at first sight, it would appear that for α ⩾ 4.6 (the point where the stationary state becomes bistable), the entire s = 0 line is a phase separation line. However, by zooming in a vertical strip of size 10−7 in this region (see Fig. 6), we find that the line separating the phases is not perfectly vertical but crosses s = 0 at α ≈ 6.6, which corresponds roughly to the transition point for the stationary state. Moreover, on this scale, it is clear that we deal with a steep but smooth transition between phases.

FIG. 6.

Phase boundaries at the s = 0, α ≈ 6.66 crossover with Nex = 75, 100, and 125.

FIG. 6.

Phase boundaries at the s = 0, α ≈ 6.66 crossover with Nex = 75, 100, and 125.

Close modal

Figure 6 shows that the phase separation lines become sharper with larger Nex, and a “true” phase transition emerges in the infinite pumping rate limit. A similar conclusion can be drawn by plotting the rescaled stationary mean ⟨N⟩/Nex; cf. Fig. 7. This can be intuitively understood by appealing to the effective potential (19). As Nex increases, the potential barrier becomes larger and two stable phases emerge at the point where the local minima are equal. Indeed, Fig. 3 shows the plot of the rescaled potential U/Nex as a function of the rescaled variable x = n/Nex, which approaches the (Nex independent) limit

v(x)=0xlogνν+1+sin2(αy)(ν+1)ydy

as it can be deduced from formulas (12) and (19). Therefore, in the limit of a large pumping rate, we deal with a particle in a fixed potential v(x) at inverse temperature 1/kT = Nex. At α ≈ 1, the dependence of the mean on Nex switches from constant to linear behavior as the minimum of the potential v(x) moves away from zero. When the two minima are at different heights, the lower one becomes the stable and other one is metastable. Communication between the phases becomes increasingly unlikely, with the probability decreasing exponentially with Nex. When the two minima are equal, we have two stable states, and the corresponding value of α is the phase transition point for the mean photon number.

FIG. 7.

Rescaled stationary state mean photon numbers, ⟨ρss⟩/Nex for increasing Nex, showing phase transition becomes sharp as Nex.

FIG. 7.

Rescaled stationary state mean photon numbers, ⟨ρss⟩/Nex for increasing Nex, showing phase transition becomes sharp as Nex.

Close modal

More information about the dynamical phase transitions may be obtained from the rest of the spectrum of the semigroup generator, and for a more in-depth treatment of these numerical aspects, we refer the reader to Ref. 48.

We have studied the counting process associated with the measurement of the outgoing atoms in the atom maser and shown that this process satisfies the large deviations principle. In particular, this means that the crossover behavior observed in numerical simulations is not associated with the non-analyticity of the limiting log-moment generating function, as one would expect for a genuine phase transition. The rescaled counting process Λt/Nex does, however, appear to exhibit such a transition in the limit of infinite rate Nex, as argued in Sec. IV using the potential model and illustrated in Figs. 3, 6, and 7.

As a corollary, we have showed that the counting process satisfies the central limit theorem, which can be used to develop the statistical estimation theory of local asymptotic normality.49–51 

The model we have investigated has the property that the stationary state is diagonal in the Fock basis, and all the jump operators leave the set of diagonal states invariant. The proof of the large deviations principle for the non-Markov counting process Λt relies on the quantum semigroup’s restriction to the diagonal algebra, which results in a classical birth–death semigroup (when proving the strict positivity and immediate compactness of Ts(d) and when applying Krein–Rutman theory). We leave for a future investigation the study of the same problem in settings where no classical reduction is possible; an example would be the atom maser where the outgoing atoms are measured in a different basis than the standard one, thus breaking the invariance of the diagonal algebra.

The compactness of the Markov semigroup makes our model tractable as it becomes essentially finite dimensional, as the bath decay dominates the absorption due to the atom interaction. An interesting problem would be to explore more general classes of infinite-dimensional systems (e.g., continuous variables or infinite spin chains) where a similar phenomenon holds. Another issue is the general relation between the “static” transitions, which refer to non-analytic properties of the stationary state, and dynamic transitions, which characterize properties of the measurement process. As shown in Ref. 52, one can construct examples where the stationary state does not change, while the system undergoes a dynamical phase transition.

Finally, it would be interesting to consider a more general large deviations setup, which takes into account the correlations between the detection events rather than looking at the total number of counts.

M.v.H. and M.G. thank J. P. Garrahan and I. Lesanovsky for numerous fruitful discussions; their research was supported by the EPSRC under Grant Nos. EP/J009776/1 and EP/T022140/1 and the Fellowship under Grant No. EP/E052290/1. R.C. and F.G. acknowledge the support of the INDAM GNAMPA Project 2020 “Evoluzioni markoviane quantistiche” and of the Italian Ministry of Education, University and Research (MIUR) for the Dipartimenti di Eccellenza Program (2018–2022)—Dept. of Mathematics “F. Casorati,” University of Pavia.

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Proof of Proposition 2.

  1. Since G is negative by (9), it is the generator of a strongly continuous contraction semigroup on h. We have
    D(G)D(Li)i=1,,4
    and
    Gu,u+u,Gu+i=04Liu,Liu=0uD(G).
    Consequently, L generates a minimal sub-Markov quantum dynamical semigroup by Ref. 36, Theorem 3.21. In particular, L(0) generates the sub-Markov quantum dynamical semigroup that we formally denoted by (etL(0))t0.
    In order to prove conservativity (which implies uniqueness by Ref. 36, Corollary 3.23), we can use Corollary 3.41 of Ref. 36, p. 73 with C = Φ = −2G and D = D(G); we just need to check condition (3.41) of Ref. 36, Corollary 3.41, p. 73, which is equivalent to the existence of a constant b such that for every uD(G),
    4Gu,Gu+i=042GLiu,2GLiub2Gu2.
    (A1)
    Let S be the shift operator, which acts as S|en⟩ = |en+1⟩ and whose adjoint is S*|en⟩ = δn⩾1|en−1⟩; note that a=S*N and a*=NS. We recall some relations that are used in the following computation:
    aa*=N+1,a*a=N,
    Sf(N)|en=f(n)|en+1=f(N1)S|en
    for any suitable function f of the number operator. We have
    4G22i=04Li*GLi=(Nex+ν+N(2ν+1))2+Nexsin(ϕN+1)S*(Nex+ν+N(2ν+1))Ssin(ϕN+1)+Nexcos2(ϕN+1)(Nex+ν+N(2ν+1))+(ν+1)a*(Nex+ν+N(2ν+1))a+νa(Nex+ν+N(2ν+1))a*=(Nex+ν+N(2ν+1))2+Nexsin2(ϕN+1)(Nex+ν+(N+1)(2ν+1))+Nexcos2(ϕN+1)(Nex+ν+N(2ν+1))+(ν+1)N(Nex+ν+(N1)(2ν+1))+ν(Nex+ν+(N+1)(2ν+1))(N+1)=0N2(2ν+1)N+B,
    where B is a bounded operator. Since −2G = (Nex + ν) + (2ν + 1)N, we obtain (A1).
    Equation (11) follows from Ref. 36, Propositions 3.18; by the definition of a minimal quantum dynamical semigroup given by Fagnola and Chebotarev, we have that T is approximated in the pointwise w*-topology by the following maps: for every XB(h), u, vD(N),
    u,T(0)(t)(X)v=etGu,XetGv,u,T(n+1)(t)(X)v=etGu,XetGv+i=140tLiesGu,T(ts)(n)(X)LiesGvds.
    (A2)
    For every n ⩾ 0, Eq. (A2) extends uniquely to a bounded bilinear form represented by a bounded operator T(n)(t)(X). In our case, for every uh and s > 0, esGuD(N)n=1D(Nn) and Li(D(N)) ⊂ D(N); hence, the expressions in Eq. (A2) make sense for every u,vh (the function that is inside the integral is well defined for every u,vh unless when s = 0, which is a set with zero Lebesgue measure). Moreover, we can get by recursion the following explicit expression for u,T(n+1)(t)(X)v, u,vh:
    u,T(n+1)(t)(X)v=etGu,XetGv+k=1ni1,,in=140t1tktf(t,u,v,X;t1,i1,,tk,ik)dt1dtk,
    (A3)
    where
    f(t,u,v,X;t1,i1,,tk,ik)=eG(ttk)LikeG(t2t1)Li1eGt1u,XeG(ttk)LikeG(t2t1)Li1eGt1v.
    Since for every t ⩾ 0, Tt(n)(X) converges to Tt(X) in the w*-topology (monotonically for positive X), we conclude.
  2. 2. and 3. By restricting Tt to diagonal operators in the Fock basis, we obtain the semigroup of a birth–death process [see point (iii) in Sec. III for the details]. We can, then, apply standard arguments (Ref. 44, Sec. V D) to see that ρss is a faithful normal invariant state. This implies that the semigroup is positive recurrent53 and, consequently, the fixed points F(T) are contained in the decoherence-free algebra N(T). Moreover, thanks to Ref. 54, Theorem 3.2, N(T) is trivial [i.e., N(T)=C1]. Then, 1 is the unique harmonic projection and so ρss is the unique invariant state; finally, the semigroup is ergodic by Ref. 53, Theorem 3.3.□

Proof of Proposition 3.

  1. and 2. We fix sR, and we drop the index s for the rest of the proof. We can proceed similarly as in the Proof of Proposition 2 and use again (Ref. 36, Theorem 3.22) choosing
    L1=es/2L1,Lk=Lkfor k=2,3,4,2G=k=14Lk*Lk+(es1)1s>01+1νNex1=2G+(es1)1s>01+1νNex1.
    For the purposes of this proof, it would be enough to define −2G′ as −2G + (es − 1)1s>0Nex1; the extra factor 1+1ν is required in order to keep the same notation in the Proof of Lemma 7.

    Note that the following holds:

    • G′ generates a strongly continuous contraction semigroup on h.

    • For all uD(G′) = D(N),

    Gu,u+u,Gu+k=14Lku,Lku0.
    Then, the operator L, defined as
    L(X)={G,X}+k=14Jk(X)=L(s)(X)+(1es)1s>01+1νNexX,
    generates a (sub-Markovian) quantum dynamical semigroup T, which is the minimal solution to
    u,T(t)(X)v=etGu,XetGv+i=140tLierGu,Ttr(X)LierGvdr
    (A4)
    for every u, vD(G′) = D(G) and which is approximated by the sequence of maps
    u,T(0)(t)(X)v=etGu,XetGv,u,T(n+1)(t)(X)v=etGu,XetGv+i=140tLierGu,Ttr(n)(X)LierGvdr
    for u,vh. Note that when s ⩽ 0, G = G′; hence, we can take TsT, while when s > 0, we can take Ts(t)=et(es1)1+1νNexT(t). Hence, again, the conclusion immediately follows.
    Let us prove the uniqueness of the solution to Eq. (16) and that its w*-infinitesimal generator (Ls,D(Ls)) is equal to (L+(es1)J1,D(L)). Let us consider T̃ the semigroup generated by (Ls+C,D(Ls)) (see Theorem 12 in  Appendix B), where C is the bounded completely positive linear map defined as
    C(X)=(1es)J1(X)+1s>0Nex(es1)X,XB(h).
    (A5)
    Because of Corollary 13 in  Appendix B, T̃ is again a w*-continuous quantum dynamical semigroup. Note that Eq. (B1) implies
    T̃(t)(X)=T̃s(t)(X)+0tT̃(tr)CT̃(r)(X)dr,
    (A6)
    and differentiating Eq. (A6) multiplied by etNex(1es)1{s>0}, we obtain that etNex(1es)1{s>0}T̃ solves for
    ddtu,etNex(1es)1{s>0}T̃(t)(X)v=L*(|vu|)(etNex(1es)1{s>0}T̃(t)(X))
    (A7)
    for u, vD(G). Since Eq. (A7) admits a unique w*-continuous positive solution (see Ref. 36, Corollary 3.23), etNex(1es)1{s>0}T̃ must coincide with T and so do their w*-infinitesimal generators, which are, respectively, (Ls+(1es)J1,D(Ls)) and (L,D(L)). Hence, we get that (Ls,D(Ls)) is equal to (L+(es1)J1,D(L)) and the solution of Eq. (16) is, indeed, unique.
  2. By the previous arguments, we have, following once again the same line of the Proof of Proposition 2, the integral representation for the semigroup T. When s > 0, we just have to introduce the correcting multiplicative term.□

Theorem 12
(Theorem 3.1.33, p. 191, Ref. 37). Let S be the generator of a σ-continuous semigroup (P(t))t0, with σ equal to either the weak or the weak* topology. If C is a bounded and σσ-continuous, then (S+C) generates a σ-continuous semigroup P(S+C) of bounded operators, and for every t ⩾ 0, XB(h),
P(S+C)(t)(X)=P(t)(x)+k10t1tktP(t1)CP(t2t1)CP(tktk1)CP(ttk)(X)dt1dtk.
(B1)
The integrals define a series of bounded operators that converges in norm; the integrals are defined in the norm topology when σ = w and in the weak* topology when σ = w*.

Corollary 13.

In the conditions of the previous theorem, if we additionally suppose that both C and the maps P(t) are completely positive, then also the perturbed semigroup P(S+C) is completely positive.

Proof.
Recall that the composition of two completely positive (c.p.) maps is still c.p. Then, all the integrands in the integral form before are c.p. Now, remember another equivalent characterization of c.p.: Φ on B(h) is c.p. iff for any nN and for any X1, …, Xn and Y1, …, Yn bounded operators,
i,j=1nXi*Φ(Yi*Yj)Xj0.
Then, it is immediate to see that
i,j=1nXi*P(S+C)(t)(Yi*Yj)Xj=i,j=1nXi*P(t)(Yi*Yj)Xj+k10t1tkti,j=1nXi*P(t1)CP(t2t1)CP(ttk)(Yi*Yj)Xjdt1dtk0.

Proposition 14.

Let B be a Banach space and (S(t))t⩾0 be a strongly continuous semigroup with generator [A, D(A)]. If MB is such that

  1. M is dense in B,

  2. M is a set of analytic vectors for A, that is, Mn0D(An), and for every XM,
    n=0+znn!An(X)
    has a positive radius of convergence, and
  3. A(M)M,

then M is a core for A.

Proof.

By the definition of core, we need to prove that M is dense in D(A) with respect to the graph norm ‖XA ≔ ‖X‖ + ‖A(X)‖. Fix XD(A), ϵ > 0. We shall consider successive approximations of X.

Step 1.
limt0X1t0tS(u)(X)duA=0.
Note that due to the strong continuity of S,
limt0X1t0tS(u)(X)du=0
is trivial, and note that
limt0A(X)1tA0tS(u)(X)du=limt0A(X)S(t)(X)Xt=0
follows from the definition of infinitesimal generator. Therefore, there exists t̃>0 such that
X1t̃0t̃S(u)(X)duAϵ.
Step 2. There exists YM such that
1t̃0t̃S(u)(X)du1t̃0t̃S(u)(Y)duA<ϵ.
Indeed, M is dense in B; hence, we can find YM such that XY<ϵt̃/4; we have
1t̃0t̃S(u)(X)du1t̃0t̃S(u)(Y)du1t̃0t̃S(u)(XY)duϵ/2
and
1t̃A0t̃S(u)(X)du1t̃A0t̃S(u)(Y)du=1t̃St̃(XY)+(XY)ϵ/2,
and summing up, we conclude.
Step 3. If we assume that t̄ is small enough, there exists N > 0 such that
1t̃0t̃S(u)(Y)du1t̃0t̃k=0NukAk(Y)k!du<ϵ/2
and, since A(Y)M too,
1t̃0t̃S(u)(A(Y))du1t̃0t̃k=0NukAk(A(y))k!du<ϵ/2.
Hence,
1t̃0t̃S(u)(Y)du1t̃0t̃k=0NukAk(Y)k!duA<ϵ/2,
and note that
1t̃0t̃k=0NukAk(Y)k!du=k=0N1t̃0t̃ukk!duAk(Y)M.
Hence, we have shown that for every XD(A) and for every ϵ > 0, there exists an element in M, which is far from X less than 4ϵ with respect to ‖·‖A, and we are done.□

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