Time-separated entangled light pulses from a single-atom emitter

David Vitali¹, Priscilla Cañizares¹, Jürgen Eschner² and Giovanna Morigi³

¹ Dipartimento di Fisica, Università di Camerino, 62032 Camerino, Italy
² ICFO—Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
³ Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain

E-mail: Giovanna.Morigi@uab.es

Received 20 December 2007
Published 17 March 2008

Contents

• 1. Introduction
• 2. Theoretical description
  • 2.1. The system
  • 2.2. Quantum Langevin equations
    • 2.2.1. Dynamics during the first laser pulse
    • 2.2.2. Dynamics between the two laser pulses
    • 2.2.3. Dynamics during the second laser pulse
• 3. Quantifying the entanglement between the two pulses
• 4. Results
• 5. Conclusions
• Acknowledgments
• Appendix A. Mechanical effects in the Langevin force
• Appendix B. Effective quantum Langevin equations
  • B.1. First laser pulse
  • B.2. Second laser pulse
• Appendix C. Calculation of the elements of the correlation matrix
• References

1. Introduction

The increasing interest in realizing quantum networks by means of atoms and photons has raised the issue of achieving full coherent control of atom–photon interactions. Photonic interfaces with atomic ensembles, namely, with a macroscopic number of atoms, have been explored in several milestone experiments, which demonstrated the generation of single photon sources [1], two-mode squeezing in the polarization of emitted light [2], atomic memory for quantum states of light [3]–[5], entanglement of remote ensembles [6] and teleportation between light and matter [7].

Complementary to this approach, photonic interfaces using single atoms can take advantage of the large level of control that can be achieved on the atomic internal and external degrees of freedom. For instance, in microwave cavity QED, quantum state and entanglement engineering using single atoms have been demonstrated [8, 9]. In the optical regime, milestone experiments demonstrated the realization of a one-atom laser [10, 11], revealed the mechanical forces of single photons on single atoms [12]–[14], achieved the controlled interaction of a trapped ion and a cavity [15, 16], yielding single-photon generation on demand [17]–[19], and characterized the entanglement between a single atom and its emitted photon [20, 21]. This latter step was instrumental for establishing entanglement between two distant trapped particles by projective measurement of the emitted photon [22]. Most recently, reversible quantum state transfer between light and atoms in a cavity has been experimentally demonstrated [23]. These results constitute relevant progress towards the realization of quantum networks with single atoms [24, 25].

A possible implementation of quantum networks with continuous variables [26] using single atoms as interfaces requires the controlled interaction of light with the atomic external degrees of freedom, which exploits the mechanical effects of light–atom interactions [27]–[29]. Using these concepts, in a recent proposal we predicted that a single atom, confined inside a resonator, can act as a quantum medium, generating quantum correlations in the emitted light [30, 31]. Such correlations can be two-mode squeezing types of correlations, which for bipartite Gaussian systems are synonymous of EPR-entanglement [26, 32]. In particular, under suitable conditions, two classical light (laser) pulses, temporally separated at the input, exhibit two-mode squeezing types of correlations at the output of this kind of device, as sketched in figure 1. In this case, the quantum state of the atomic motion serves as intermediate memory which mediates the entanglement between the first and second pulses at the cavity output. Variation of the laser parameters driving the atom allows the degree of entanglement between the pulses to be tuned.

In this work, we analyze the efficiency of the proposal for temporally separated entangled pulses with single atoms [30, 31] by using a quantum Langevin equation description. This description permits us to determine the correlation matrix and hence the amount of entanglement one obtains using experimentally accessible parameter regimes. We show that this proposal is viable to existing experiments, hence providing an important step towards continuous-variable photonic interfaces with single atoms. We remark that here the atom acts as a source of continuous variable (CV) `time-bin' entangled pulses, which could be an alternative solution for secure communication [33] with respect to those employing single-photon qubits ([34] and references therein).

This paper is organized as follows. In section 2, the model determining the system's dynamics is introduced, and the quantum Langevin equations for the coupled dynamics between atomic motion and the cavity mode are derived. In section 3, the corresponding correlation matrix for the two propagating correlated pulses is determined and the degree of entanglement characterized. In section 4, entanglement is discussed as a function of the experimental parameters. The conclusions and outlook are presented in section 5, and the appendices report details of the derivations in section 2 and section 3.

2. Theoretical description

We summarize the basic concepts at the basis of the proposal for the creation of pairs of temporally separated, CV entangled pulses. Let us consider a single atom in a harmonic trap of frequency ν and confined inside a resonator, in the set-up sketched in figure 1. Let b, b^† be the annihilation and creation operator of an excitation of the quantized motion inside the trap. The atomic dipole couples with a cavity mode at frequency ω_c, which is far-off resonance from the dipole frequency ω₀. We denote by a, a^† the annihilation and creation operator of a cavity photon. A first laser pulse at frequency ω_L≊ω_c + ν illuminates the atom in the time interval [0, T₁]. In this regime, the simultaneous emission of a cavity photon and of a vibrational quantum of the atom is resonantly enhanced, see figure 2(a), and the relevant dynamics are described by the effective Hamiltonian

where χ₁ is the transition amplitude of the resonant process. This Hamiltonian describes an interaction giving rise to two-mode squeezing, i.e. CV entanglement between the center-of-mass oscillator and the cavity mode. If the pulse is implemented for a sufficiently short time, so that photon leakage out of the cavity has negligible effects and the dynamics can be assumed to be coherent, at the end of the pulse the cavity mode and the atom's vibrational motion will be entangled. Cavity decay will give rise to a propagating pulse, whose photon number is quantum-correlated with the vibrational phonon number [28, 29].

Figure 2. Relevant energy level and resonant transitions during the two pulses. Here, |g and |e denote ground and excited states of the atomic dipole transition with linewidth γ, nc the number of cavity photons, and nmec the number of vibrational excitations. (a) First pulse: the laser is detuned by Δ1 from the dipole, and by δ1 = ν from the cavity mode, driving resonantly the two-photon transition |g, nmec, nc→ |g, nmec + 1, nc + 1. These processes generate two-mode squeezing between the cavity and the motion. (b) Second pulse: the laser is detuned by Δ2 = Δ1–2ν from the dipole, hence driving resonantly the two-photon transition |g, nmec, nc→ |g, nmec–1, nc + 1. Choosing the pulse duration properly, quantum-state transfer between motion and cavity mode can be achieved, see [27, 28]. Note that by varying the frequency of the second laser pulse and/or the atomic transition which it excites, the two output pulses can be at different frequencies or wavelengths.

We then assume that at a time T > T₁ a second laser pulse tuned to the frequency ω_L≊ω_c–ν drives the atom until the time T₂. In this regime, one has resonant enhancement of the emission of a cavity photon with the simultaneous absorption of a vibrational quantum, see figure 2(b). The effective dynamics is described by the Hamiltonian

where χ₂ is the transition amplitude of the resonant process. Again, we assume that cavity decay can be safely neglected, so that the dynamics is coherent. When the pulse duration δT = T₂–T is appropriately chosen, the quantum state of the center-of-mass motion at time T is completely transferred to the cavity mode at time T₂. Consequently, by cavity decay a second pulse at the cavity output will be generated, which is entangled with the first one [30, 31].

This proposal is based on the assumption that during the laser pulses the dynamics is essentially described by the effective Hamiltonian (1) and (2), while detrimental effects like atomic spontaneous emission, vacuum optical input noise entering the cavity and fluctuations of the trapping potential can be neglected. These assumptions are justified in certain parameter regimes, which have been discussed in [30, 31]. The scope of this work is to explore the robustness of the scheme when detrimental effects are small but cannot be a priori neglected in the dynamical equations. For this purpose, in this section we adopt a quantum Langevin equations treatment, taking into account all sources of noise. This permits us to determine the correlation matrix for the generated pulses, which are discussed in section 3, and to quantify their degree of entanglement for a wide range experimental parameters, as shown in section 4.

2.1. The system

We consider an atom of mass m, whose center-of-mass motion takes place essentially in one-dimension (1D). We assume, in fact, that the radial potential is sufficiently steep, so that the radial motion can be considered frozen out. Let the motion be along the -axis, and let x, p be the position and momentum of the atomic center-of-mass. The center-of-mass is a harmonic oscillator with angular frequency ν, whose Hamiltonian reads

where x, p are related to the annihilation and creation operators b and b^† of a quantum of vibrational energy ν by the relations and . The relevant atomic internal degrees of freedom are the ground state |g and the excited state |e, which form a dipole transition with moment d and frequency ω₀. The transition couples to one optical mode of the cavity at frequency ω_c and to the laser, a classical field, which is a pulsed excitation whose central frequency ω_L is tuned from one pulse to the next. We define our model in the reference frame rotating at ω_L, remembering that the frequency changes from the first to the second pulse, and the two reference frames are hence related by a global time-dependent phase. In the reference frame rotating at ω_L the total Hamiltonian is H = H_a + H_c + H_int. Here,

is the atomic Hamiltonian with Δ = ω_L–ω₀, while

describes the dynamics of the cavity mode, with δ = ω_L–ω_c. The coupling between the atom and the fields is described by

where σ = |ge| and σ^† = |eg| denote the dipole lowering and raising operators and B(t) is the operator for the field degrees of freedom, which we decompose into laser and cavity components, B(t) = B_L(t) + B_c. The cavity term is B_c = g_c cos(kx cos θ_c + _c)a, where g_c is the coupling strength and the cavity mode wave vector ( ) forms an angle θ_c with the axis of the motion. The angle _c takes into account the position of the trap center inside the cavity. From now on we will assume that the atomic motion is in the Lamb–Dicke regime, such that the atom–photon interactions can be expanded at second order in the Lamb–Dicke parameter . In this limit term B_c takes the form^Note4 

The laser term is B_L(t) = Ω(t)e^ikx cosθ_L, where Ω(t) is the (slowly varying) Rabi frequency and θ_L is the angle between the direction of propagation of the laser and the trap axis. In the Lamb–Dicke regime, this operator reads

2.2. Quantum Langevin equations

The full dynamics of the system must take into account the coupling with the environment, which is here represented by the dipole fluctuations, giving rise to spontaneous emission, by vacuum fluctuations at the cavity input, giving rise to cavity decay, and by fluctuations of the trapping potentials, which are responsible for damping and loss of quantum coherence of the center-of-mass motion. Cavity decay is described by the Markovian noise operator aⁱⁿ(t), while the heating due to the fluctuations of the trap potential is described by the phenomenological Markovian input noise operator bⁱⁿ(t) acting on the atomic motion. These two noise sources are mutually uncorrelated and have zero mean value, and their only nonzero second-order correlation functions are

where is the mean thermal vibrational number of the effective thermal reservoir coupling to the atom center-of-mass motion. We denote by κ_b the damping rate of the vibrational motion. Pure heating corresponds to the limit of κ_b→0, simultaneously with an infinite temperature of the associated effective reservoir, i.e. , with , the heating rate, kept constant [37].

The input noise terms associated with spontaneous emission are more involved because the latter affects both the internal and the motional degree of freedom of the atom, due to the presence of recoil. This effect has been neglected in the analysis presented in [35], which focused on the cw-generation of entangled light, and will be systematically taken into account in this work. For this purpose, the corresponding Langevin force must be defined. At second order in the Lamb–Dicke parameter η, the Langevin force operator F(t) associated with spontaneous emission is given by (see also its detailed derivation in appendix A)

where γ is the spontaneous emission rate of level |e, and is the dipole pattern of emission, such that . The integral is over the angle θ between the wave vector of the emitted photon and the axis of the motion, and f_θⁱⁿ(t) are the zero-mean, angle-dependent Langevin forces, with the only nonzero correlation function

It is convenient to isolate the zeroth-order term in the Lamb–Dicke parameter in equation (12) and to rewrite F(t) as

with the zeroth-order term

possessing the correlation function

The term F_nl(t) = F(t)–fⁱⁿ(t) is hence at higher order in η. We shall see that spontaneous emission noise is essentially described by fⁱⁿ(t), because the higher order term F_nl(t) gives a negligible contribution to the Langevin equations in the parameter regime considered in this work (see appendix B).

In the frame rotating at the laser frequency the quantum Heisenberg–Langevin equations (HLE) of the system read [38]

where σ_z = σ^†σ–σσ^† and κ is the decay rate of the cavity mode.

In the cases we are going to consider, both lasers as well as the cavity mode are tuned far-off resonance from the dipole transition, i.e. |Δ|  Ω,g, |δ|, γ. In this regime, the atomic internal degrees of freedom can be eliminated in the second-order perturbation theory. Hence, we neglect the time evolution of σ_z, equation (20), and approximate σ_z(t)≈–1. Correspondingly, equation (19) becomes

whose formal solution is

We insert solution (22) into the other HLE and neglect the transient term as we are interested in the dynamics at times which are much larger than 1/|Δ|. The resulting HLE for cavity and trap oscillators are

where we have not taken care of operator ordering, since, as we shall see, within the validity limit of our treatment these integral terms will generate only linear contributions. We now determine the solutions of these equations for the dynamics during the first pulse, between the pulses, and during the second pulse.

2.2.1. Dynamics during the first laser pulse We first consider the dynamics during the first pulse, i.e. in the time interval 0≤t≤T₁, where we assume a square laser pulse with central angular frequency ω_L1 and constant Rabi frequency Ω(t) = Ω₁ during this time interval. Correspondingly, we denote by Δ = Δ₁ and δ = δ₁ the detuning of the laser from the atomic and cavity frequency. We assume that the laser is far-off resonance from the atomic transition, i.e. Δ₁ is negative and |Δ₁|  Ω₁,g, γ. The laser frequency is tuned to the value ω_L1≊ω_c + ν, namely, the cavity mode is resonant with the Stokes motional sideband of the laser light. This condition allows to establish a parametric-amplifier type of interaction between the cavity mode and motion, which is selectively enhanced provided that the Stokes sideband is spectrally resolved, namely, when νT₁  1 and ν  κ, κ_b. In particular, we take into account the presence of ac-Stark shifts δν, due to the mechanical coupling with laser and cavity modes, by tuning the laser frequency to the value

and hence δ₁ ' = ν ', with ν ' = ν + δν and δ₁ ' = δ₁–δ ', and δ ' accounts for possible ac-Stark shifts due to off-resonant couplings. The value of δν is determined in a self-consistent way, which is extensively discussed in appendix B, see also [35].

Starting from equations (23) and (24), in this parameter regime we derive the effective HLE, which describe the coherent interaction between the cavity mode and the vibrational motion during the first laser pulse, in the presence of losses and noise processes due to spontaneous emission, cavity decay and vibrational heating. For later convenience, we study the equations in the reference frame rotating at the cavity-mode frequency, which is obtained from the reference frame of the laser frequency by the transformation

In this reference frame, the HLE read

The equations are second order in the Lamb–Dicke parameter, and have been obtained by neglecting off-resonant terms. The steps of the derivation are reported in appendix B.

We now define and discuss each term appearing in the equations. The effective coupling between motion and cavity mode is

and corresponds to the Raman processes in which laser photons are scattered into the cavity mode with a change in the center-of-mass excitation. The new noise operators appearing in equations (28) and (29), , are defined as

and describe the coupling with external optical modes due to the photons scattered by the atom. They possess the same correlation functions of the spontaneous emission noise fⁱⁿ(t), and at the time scales of interest, ν  't  1, they are uncorrelated from each other. These noise components affect both the cavity mode and the vibrational motion. In particular, describes quantum noise associated with incoherent scattering by the atom of a cavity photon into the external modes at rate

while the corresponding input noise scales with

where . The two noise terms in equations (31) and (32) also affect the atom's motion due to the mechanical effects of the scattering of laser photons. The incoherent change of vibrational quanta due to scattering of laser photons takes place at rates^Note5 

while the corresponding terms scaling the input noise are given by (see appendix B)

so that . The formal solution of equations (28) and (29) at time t = T₁ reads

where we have introduced the time-dependent functions

and the parameters

Equations (37) and (38) describe the dynamics of the coupled motion and cavity mode during the first pulse in the presence of quantum noise. By setting all noise terms to zero, κ, κ_j = 0, they reproduce the well-known coherent two-mode squeezing dynamics [38], where entanglement monotonically increases as a function of the interaction time T₁. The presence of quantum noise sets a limit to the establishing of these dynamics.

2.2.2. Dynamics between the two laser pulses In the time interval, T₁≤t≤T, the laser is turned off and consequently there are no resonant photon scattering processes which couple the cavity mode and the atom's vibrational motion. The HLE describing the system without laser excitation can be obtained immediately from equations (28) and (29) by setting Ω₁ = 0 and thus . The resulting HLE are given by

where off-resonant coupling between motion and cavity mode is neglected, and δ₁^b is defined in appendix B. This latter term is due to the definition of the reference frame rotating at frequency ν  ', which compensates the laser-induced ac-Stark shift on the motion: in the absence of the laser, this component of the frequency ν  ' is unbalanced. The assumption of neglecting off-resonant coupling between motion and cavity mode is justified when the cavity-mode wave vector is orthogonal to the axis of the motion, and thus there is no mechanical coupling. In general, it is valid at zero order in the expansion in the small parameter ηg cosθ_c/|ω_c–ω₀|, which hence imposes a condition on the integration time T under which equations (44) and (45) are valid. The solutions of equations (44) and (45) are

where and are given by equations (37) and (38). They yield the observables of interest at time T, before the second laser pulse is switched on.

2.2.3. Dynamics during the second laser pulse We now consider that in the time interval T≤t≤T₂ a square pulse of constant Rabi frequency Ω(t) = Ω₂ and central angular frequency ω_L2 illuminates the atom. We denote by δ₂ = ω_L2–ω_c, and Δ₂ = ω_L2–ω₀ the detuning of the laser frequency from the cavity mode and the dipole transition, respectively. In the limit in which processes where absorption of a laser photon and of a phonon are resonant with emission of a cavity photon, ω_L2≈ω_c–ν, we derive the HLE from equations (17)–(20), describing the interaction between the cavity mode and the vibrational motion during the second laser pulse, in the presence of losses and noise processes due to spontaneous emission, cavity decay, and vibrational heating,

where parameters χ₂ and κ^b_±2 are found from χ₁ and κ^b_±1 by replacing Ω₁→Ω₂, δ₁→δ₂, and Δ₁→Δ₂ in equations (30) and (33). The details of the derivation are reported in appendix B.

The solutions of equations (48) and (49) at the end of the second pulse, t = T₂, read

where we have introduced

and

Equations (50) and (51) give the cavity mode and motion at the end of the second pulse. Setting all decay and noise sources to zero, we recover the ideal polariton dynamics during the second pulse, namely a periodic dynamics at frequency |χ₂|. Ideally, then, the states of the motion and of the cavity mode are swapped when T₂–T = δT₂⁰, such that |χ₂|δT₂⁰ = (2ℓ + 1)π/2, with ℓ an integer. At these values the function f₂ reaches its maximum, f₂(δT₂⁰)|_{κ: j = 0} = 1, while g_–2 vanishes, g_–2(δT₂⁰)|_{κ: j = 0} = 0. The effect of decay and noise is to damp the oscillators, hence to modify the oscillation frequency and the behavior of functions f₂, g_–2. In particular, the maximum value of f₂(t) is always smaller than unity, and the maxima of f₂ do not occur at the same instants of time in which g_–2(t) vanishes. We optimize the process by setting T₂–T such that it fulfills the condition g_–2(T₂–T) = 0. Denoting by δT₂^opt = T₂–T the time interval fulfilling this condition, it satisfies the relation

3. Quantifying the entanglement between the two pulses

The two time-separated pulses at the cavity output can be considered as two independent modes, even if they originate from the same intracavity field at different times. In fact, the output field a^out(t) is related to the intracavity field by the input–output relation [38]

and it is characterized by the commutation relation . In order to have a quantity directly related to the detected field, we define the following integrated output field over a generic measurement time T_m [40],

The field operators form a class of dimensionless bosonic operators, , and they commute, i.e. they describe independent modes, as soon as they do not temporally overlap, that is, whenever |t–t ' | > T_m. The two output pulses we are interested in are therefore those associated with the operators and . In the present scheme the two pulses are temporally separated and therefore it is natural to consider T_m < T–T₁, which automatically warrants the independence of the two integrated output modes.

In order to characterize the entanglement between the two pulses, one usually considers the amplitude and phase quadratures of the two independent modes, which are in this case

and construct the correlation matrix

where we have defined the 4D vector

In equation (62), the averaging corresponds to taking expectation values with respect to the initial state of the system and the environment. We now proceed in determining its elements.

We use the definition in equation (59), the input–output relation of equation (58), and the explicit solution for in the two relevant time intervals, T_j≤t≤T_j + T_m, j = 1, 2, which is given by equation (46), and get the following expression of the integrated output fields as a function of the intracavity fields at the end of the pulses and of the input noise:

with the factor

and the input noise term

Using equation (63) in equation (62), we find that the correlation matrix V^out can be decomposed into the sum of three contributions,

where Vⁱⁿ is the contribution due to the input noise term nⁱⁿ(T_j, T_m), V^mix is the contribution due to the correlation between the intracavity fields at the end of the pulses and the input noise, and V is the correlation matrix for the quadratures of the intracavity fields , . Its elements have the form

where we have defined the 4D vector ξ^T = (X(T₁), P(T₁), X(T₂), P(T₂)), and are reported in appendix C. Using the correlation functions of the input noise and one finds that Vⁱⁿ is proportional to the 4×4 identity matrix

while V^mix has only four nonzero terms, which are all identical, , with

Equation (69) shows that the contributions due to correlations between intracavity fields at the end of the pulse and input noise are zero as soon as we choose the optimal transfer condition g_–2(T₂–T) = 0 of equation (57). (They are in any case negligible when (κ + κ_L)(T–T₁)  1.)

Let us now consider what is the optimal integration time T_m, such that V^out≈V, i.e.  the correlation matrix at the cavity output reproduces the correlation matrix between the intracavity fields. From equations (66) and (68), we see that T_m must be chosen such that the quantity α(T_m) is as close as possible to unity. In this case, most of the intracavity field is detected at the cavity output at the end of the pulse, and at the same time the contribution of the input noise is negligible. From equation (64) one gets that α(T_m)≤α_max, where , and it is achieved for T_m⁰≊1.25/(κ + κ_L).

In order to establish the conditions under which the two output pulses are entangled we consider the logarithmic negativity , a quantity which has been already proposed as a measure of entanglement [41]. In the continuous variable case can be defined as [42]

where

with Σ(V^out)≡det A + det B–2det C, and we have used the 2×2 block form of the correlation matrix

Therefore, a Gaussian state is entangled if and only if η^– < 1/2. This is equivalent to Simon's necessary and sufficient entanglement criterion for Gaussian states of a non-positive partial transpose [43], which can be written as 4 det V < Σ–1/4.

We finally comment on our choice to quantify the entanglement in terms of the logarithmic negativity instead of EPR variances [32]. The latter would seem a natural choice, but in fact they provide an unambiguous characterization of entanglement only for simple examples of CV two-mode entangled states, such as the two-mode squeezed state. Any reasonable entanglement measure has to be invariant under local transformations of the quadratures of each mode separately. In our system, however, the common definition of EPR correlations [42, 43] reads

with Δ(A)² the variance of A, and depends upon the chosen set of quadratures, i.e. it does not possess such invariance. The light pulses at the cavity output are in a two-mode squeezed state only in the ideal limit when the noise contributions to the dynamics are negligible, and when the ion's motional state created by the first pulse is perfectly transferred to the second pulse. Under realistic conditions the state of the two pulses is rather different from a two-mode squeezed state; therefore using EPR variances would give an ambiguous quantification of the generated entanglement. Nonetheless, there is a connection between squeezing and the measure of entanglement provided by the logarithmic negativity. To be more specific, a simple and direct quantitative connection between EPR variances and can be found in the case of symmetric bipartite states, i.e. states which are invariant under exchange of the two modes (the two pulses in our case) [42]. In such a case, the quantity gives the largest amount of EPR correlations, i.e. the minimum achievable value of ξ_EPR which can be attained in the CV bipartite state by means of local operations, i.e. by considering all possible linear combinations of the quadratures of each pulse.

4. Results

We now analyze the basic requirements and the efficiency of this scheme using parameters accessible in present experiments with atoms in resonators. Our considerations follow and extend the corresponding discussion in [31]. For comparison, we will discuss along with some realistic sets of parameters an idealized case where the cavity decay rate is set to a very small value, and the noise terms are suppressed.

The atom's internal degrees of freedom need to provide an optical dipole transition which couples to both the laser and the cavity mode. A suitable example would be an F = 0↔F ' = 1 closed atomic transition with the quantization axis along the cavity axis, and , , and laser polarization mutually orthogonal. Ideal candidates would then be alkali-earth-metal atoms or alkali-earth-metal-like ions, but other geometries can be found which allow for the realization of this scheme using also alkali atoms or alkali-like ions. We consider an alkali-earth atom in the geometrical configuration θ_L = 0, θ_c = π/2 and _c = 0. This means that the trap center coincides with an antinode of the cavity mode, and that the motion takes place along the direction of the laser beam and orthogonally to the cavity axis, such that there is no mechanical effect of the cavity field on the motion. This assumption is not strictly necessary but it is made here in order to simplify the discussion.

In order to favor motional Raman transitions over resonant scattering, we assume a value of the atom-field detuning Δ which exceeds the values of the coupling strengths Ω and (n is the average number of cavity photons) and of the atomic transition linewidth γ. Taking a typical value of the linewidth of an optical dipole transition, γ = 2π×5 MHz, we choose |Δ| = 2π×120 MHz, Ω = 2π×10 MHz and g_c = 2π×1 MHz, which are accessible values for state-of-the-art experiments with trapped atoms or ions in resonators [15, 19, 44].

We also consider the motion to be restricted to the Lamb–Dicke regime, with a Lamb–Dicke parameter η = 0.1. This leads to |χ₁|, |χ₂|≊2π×8 kHz for the coupling constants, while the loss rates associated with the various scattering processes are κ_L~κ^b_±j≊2π×0.2 kHz. The trap frequency can be set to ν≊2π×1 MHz, which is typical in ion trap experiments [36]. The heating rate of the vibrational motion may be estimated as κ_h≊2π×20 Hz [45, 46].

The value of the cavity decay rate κ must warrant the coherent creation of correlations during the pulses (κT_{1, 2}  1), as well as spectral resolution of the sidebands (κ  ν). In the examples we discuss, we assume values that range from κ≊2π×20 kHz, which is experimentally accessible [44, 47], down to 2π×1 kHz, which is more difficult to reach with the present day technology, but serves as an idealized case for comparison.

Given the parameters, the laser pulse durations T₁ and T₂–T, as well as their separation T must be adjusted in order to (i) create significant entanglement between the first pulse and the motion, and (ii) efficiently realize the quantum state transfer between the motion and the second pulses. At the same time the motion, which acts as intermediate quantum memory, needs to remain coherent during laser excitation and cavity output.

We first focus on the dependence upon the duration of the second pulse, T₂–T. The optimization of the state transfer from the center-of-mass motion to the cavity mode discussed in section 2.2.3 yielded the condition of equation (57), T₂–T = δT₂^opt. Therefore, we expect the entanglement to be maximum around this condition. This is confirmed by figure 3, where the logarithmic negativity is plotted versus the normalized duration of the second pulse, (T₂–T)/δT₂^opt at various values of the cavity decay rate. The duration of the first pulse has been fixed at T₁ = 40μs ≊2/|χ₁|^–1, while the time interval between the two pulses has been chosen to be related to the cavity decay time according to T–T₁ = 2/(κ + κ_L); the other parameter values are those discussed above. It is evident that the entanglement between the pulses is optimized when the duration of the second pulse satisfies equation (57).

Figure 3. Logarithmic negativity versus normalized duration of the second pulse (T2–T)/δT2opt (see equation (57)) at four different values of the cavity decay rate. From bottom to top, κ = 2π×(16, 11, 6.4, 0.8, 0.8) kHz; the top solid curve is plotted for comparison and corresponds to the smallest value of κ with all other noise terms set to zero. The other parameter values are (|Δ|, γ, Ω, gc, ν) = 2π×(120, 5, 10, 1, 1) MHz and η = 0.1, yielding (|χ1|, |χ2|, κL, κb±j) = 2π×(8.5, 8.5, 0.18, 0.18) kHz; the heating rate is κh = 2π×20 Hz. The duration of the first pulse is fixed at T1 = 40 μs~2/|χ1|–1, while for each curve, the time interval between the two pulses is related to the cavity decay time according to T–T1 = 2/(κ + κL) = (20, 28, 49, 327, 400) μs (bottom to top).

The dependence of the logarithmic negativity upon the time separation T–T₁ between the two pulses is shown in figure 4. Parameter values are the same as in figure 3 except that we have fixed the duration of the second pulse at the optimal value T₂–T = δT₂^opt given by equation (57). Entanglement (i.e. ) decays essentially linearly, and notably its lifetime is independent of the cavity decay rate κ. In fact, always vanishes when T–T₁≊κ_h^–1, i.e. the two pulses are entangled provided that their separation is not larger than the vibrational heating time κ_h^–1. This is not surprising because the vibrational motion acts as the CV quantum memory mediating the entanglement, and thus the heating time limits the achievable coherent storage time.

Figure 4. Logarithmic negativity versus the time separation between the two pulses T–T1 at the same values of the cavity decay rate κ as in figure 3. The other parameter values are also the same as in figure 3 except that the duration of the second pulse has been fixed at the optimal value T2–T = δT2opt = (19, 22, 24, 29, 29) μs (bottom to top).

Finally, the dependence of the logarithmic negativity upon the duration of the first pulse T₁ is shown in figure 5. The other two timing parameters are, for each curve, T₂–T = δT₂^opt as in figure 4, and T–T₁ = 2/(κ + κ_L) as in figure 3. The rest of the parameters have the same values as before. is always increasing and then tends to saturate at a value that, as expected, is larger for smaller cavity decay rates. In the presence of noise one finds an empirical expression for the asymptotic logarithmic negativity given by . This behavior can be intuitively explained by the fact that the first laser pulse entangles the cavity mode and the vibrational motion like in a parametric amplifier, and this continuous variable entanglement increases for increasing T₁. Cavity losses, however, limit the entanglement generation and are ultimately responsible for the saturation of the entanglement at larger T₁.

Figure 5. Logarithmic negativity versus duration of the first laser pulse T1, for the same values of κ as in the previous figures. The time interval between the two pulses and the duration of the second pulse are set for each curve according to T–T1 = 2/(κ + κL) and T2–T = δT2opt. The other parameter values are the same as before.

From figure 5 it can be seen that choosing a large T₁, in the saturation regime, has two advantages: (i) entanglement is maximized; and (ii) the scheme is insensitive to fluctuations of T₁. It must be kept in mind, however, that we are considering the entanglement between two output light pulses which are counted for a time interval T_m⁰ = 1.25/(κ + κ_L) starting only when each exciting laser pulse has finished (see section 3). This means that for large values of T₁ the detection of the two entangled pulses is more difficult, especially when the cavity decay rate is large. In these latter cases, the number of photons leaving the cavity during the first excitation pulse is larger than the average number of photons in the two detected output pulses. In the saturation regime T₁ = 40 μs, the average number of photons per entangled pulse is for κ = 2π×(0.8,6.4,11,16) kHz, where we have set T_m⁰ = 1.25/(κ + κ_L), while the other parameters are as in the figures.

5. Conclusions

To conclude, we have characterized the quantum correlations of two temporally separated entangled light pulses, emitted from a single atom inside an optical cavity, in the set-up first proposed in [30, 31]. By means of a quantum noise analysis we have quantified the amount of entanglement one can extract from this system for experimentally accessible parameter regimes, and we have shown that the quantum motion of a single trapped particle is an efficient quantum medium which creates and mediates entanglement on demand between subsequent radiation pulses.

This scheme offers promising perspectives for atom–photon interfaces and for devising new cryptographic schemes exploiting time-correlated pulses and continuous alphabets, thus extending those based on time-bin entangled photon pairs [34]. The scheme may also be easily generalized to the generation of two pulses of different frequencies, or even at vastly different wavelengths, by varying appropriately the frequency or wavelength of the second excitation pulse.

An interesting outlook is to study the scalability of the scheme when the number of atoms composing the quantum medium is increased in a controlled way, hence characterizing which resources the collective excitations of the medium may offer for creating entangled light.

Acknowledgments

We acknowledge discussions with Stefano Mancini. This work was partly supported by the European Commission (`CONQUEST', MRTN-CT-2003-505089; `EMALI', MRTN-CT-2006-035369; `SCALA', contract no. 015714), by the Spanish Ministerio de Educación y Ciencia (`QOIT', Consolider Ingenio 2010 CSD2006-00019; `LACSMY', FIS2004-05830; `QLIQS', FIS2005-08257; `QNLP', FIS2007-66944; Ramon-y-Cajal program) and by the Italian Ministero dell Università e Ricerca (PRIN-2005024254).

Appendix A. Mechanical effects in the Langevin force

We derive the Langevin force considering the mechanical effects of the spontaneously emitted photon on the atom. For this purpose, we consider a simple model, constituted by a dipole of frequency ω₀ at position x and Hamiltonian of the center-of-mass H_CM,

The dipole is coupled to the modes of the em-field at frequency ω_s and Hamiltonian

with interaction Hamiltonian

where g_s is the vacuum Rabi frequency. The formal solution of the Heisenberg equation for a_s gives

We substitute this result into the Heisenberg equation for σ, and obtain

We now make the Markov approximation, assuming that the characteristic frequencies of the center-of-mass motion are much smaller than the optical frequencies of the em-field, which couple quasi-resonantly with the dipole. Hence, in the integral we approximate x(t–τ)≈x(t). Taking σ(t–τ)≈e^iω₀τσ(t), we obtain

The integral gives a real term, the linewidth , and an imaginary part, the Cauchy principal value shifting the transition frequency. Including this shift in the value of ω₀ we obtain

with the noise source

such that F₀(t)F₀(t ')^† = γδ(t–t '). Let us now investigate the form of F₀(t) when the center-of-mass motion is a harmonic oscillator, H_CM = H_mec given by equation (3). By applying the Lamb–Dicke expansion, F₀(t) takes the form

with k_sx = η cosθ_s(b^† + b), where θ_s is the angle between the wave vector of the emitted photon and the axis of the motion. Taking the continuous limit of the sum, we separate the integrals over the modulus and the polar angle, and over the azimuthal angle θ = θ_s, obtaining F₀(t)≈F(t), with

where is the dipole pattern of emission, and f_θⁱⁿ(t) are the angle-dependent Langevin forces,

and which have zero mean value.

Appendix B. Effective quantum Langevin equations

B.1. First laser pulse

Starting from equations (23) and (24), which are defined in the reference frame of the laser, we move to a frame rotating at the effective vibrational angular frequency ν  ' ≊ν, and we neglect all the terms oscillating at ν  ' or larger. This approximation is justified in the regime we consider, where we assume that the Stokes and anti-Stokes sidebands are spectrally resolved. The operators in this reference frame are connected to the ones in the reference frame of the laser by the transformation , , and their equations of motion have the explicit form

where ···_θ in the equation for denotes the average over the azimuthal angle θ with weight given by the dipole pattern of emission , and we have introduced the noise operators and , which are still delta-correlated. We use the explicit expression for B(t–s), thereby neglecting the terms oscillating at ν  ' or faster, and perform the time integrals by making the Markovian approximation , for m = –1, 0, 1.

After long but straightforward calculations we get the final, effective HLE at leading order in the Lamb–Dicke parameter, which read

Here, the coefficients

correspond to the Raman processes in which laser photons are scattered into the cavity mode with a change in the center-of-mass excitation. Since we assume γ  |Δ₁| (and also ν  |Δ₁|), we shall take , and approximate χ₁ with equation (30). The noise operators are defined in equations (31) and (32), while the rates κ_L, κ_±1^b are defined in equations (33)–(35). The noise scaling factors of equation (36) are generally given by

but they reduce to the expression of equation (36) because the average over the dipole pattern gives cos θ_θ = 0.

The operators F_a and F_b in equations (B.3) and (B.4) represent nonlinear noise terms associated with incoherent scattering processes. They give rise to ac-Stark shift and losses, whose effect is, in general, detrimental for the effectiveness of the two-mode squeezing processes, and are of the form , , hermitian conjugates, and corresponding input noise operators. These terms can be neglected in comparison with the laser-induced Raman scattering processes whenever the inequality is satisfied (n is the average number of cavity photons), which is a condition on the parameters and on the geometry of the set-up. We will assume this regime, and these terms will be neglected during the laser pulses. In this regime, the frequency shifts of the cavity mode and of the vibrational motion read

These shifts must be taken into account when tuning the frequency of the first laser pulse. In particular, since the dynamics we seek relies on resonant two-photon processes, where a laser photon is absorbed and a cavity photon and a vibrational phonon are emitted, hence the cavity mode frequency must be exactly at resonance with the Stokes sideband of the driving laser, i.e.

Here, δν can be extracted from equation (B.8) when |δν|  ν, which is satisfied provided that η|Ω₁|  |Δ₁| and η|Ω₁/Δ₁|  ν, and reads

Consequently,

determines the effective frequency of the Stokes sideband.

We notice that the resonant condition (B.9), giving the relation ω_L1–ν  '  = ω_c + δ  ', gives us a simple relation between the observables in the laboratory frame and the `tilded' observables, which are connected by the relations

B.2. Second laser pulse

The main difference with the treatment of the dynamics during the first pulse is that the second pulse is set to be a different resonance condition, e.g.

where ν ''≊ν and the difference accounts for the ac-Stark shifts induced by the second laser pulse on cavity and motion frequency. When the condition (B.14) is fulfilled, the cavity mode is resonant with the anti-Stokes motional sideband of the laser light, with angular frequency ω_L2 + ν ''. Spectral resolution of this resonance is warranted when νT₂  1 and ν  κ, κ_b.

In the reference frame rotating at the frequency of the cavity mode, the HLE read

where the noise operators and are the same as in equations (B.1) and (B.2). Following the procedure outlined before, we finally get the coupled HLE

Let us now define the coefficients appearing in these equations. The quantities κ_L and δ  ' are given by equation (33) and (B.7), respectively. In fact, together with κ_L, they do not depend upon the properties of the driving laser; moreover the two noise operators and are given by equations (31) and (32). The nonlinear terms F_a and F_b are the same as in equations (B.3) and (B.4), and are negligible as we take Ω₂ cos θ_L  g_c cosθ_c. The coupling constants associated with the Raman scattering processes are

As we consider the limit γ  |Δ₂| we shall take from now on. The incoherent emission or absorption of a vibrational quantum scales with the rates

and in the limit |Δ₂|  γ, ν the rates scaling the input noise read

Finally, the frequency shift of the vibrational motion reads

For |Δ₂|  γ, η|Ω₂/Δ₂|  ν, we find with good approximation

determining, together with equation (B.7), the resonance condition for the central frequency of the laser pulse,

For |Δ₂|  ν, choosing Ω₂ = Ω₁, then δ₂^b≈δ₁^b, ν '' = ν ' and the processes leading to absorption of a phonon and emission of a cavity photon are resonantly enhanced by choosing the frequency of the second laser pulse at ω_L2 = ω_c–ν '. We consider this regime, as it substantially simplifies the calculations. When it is not fulfilled, one must consider an accumulated phase, which gives simply a total phase shift and hence modifies the quadratures exhibiting entanglement.

Appendix C. Calculation of the elements of the correlation matrix

We derive here the elements of the intracavity correlation matrix V of equation (67), assuming that the cavity modes and the vibrational motion are initially in the vacuum state, and the fact that input noise is uncorrelated with the cavity mode operators at former times. The elements read

Using equations (37), (46) and (50) we obtain

where we have used , and that κ_h  κ_b. In deriving these relations we also assumed that . In order to fully determine the above relations as a function of the initial conditions, we need

In deriving these expressions we assumed that the HLE conserve the commutation relations. This is not fulfilled for long times, when the perturbative expansion loses validity.

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Notes

Note4  Note that equations (7) and (8) are first order in the Lamb–Dicke parameter; nevertheless, we have also included the term η²(2b^†b + 1), which belongs to the second-order expansion. This term gives rise to ac-Stark shifts induced by the field, which should be systematically taken into account when one considers transition rates between vibrational states (which are at second order in η). See for instance [35, 36].

Note5  The rates in equation (35) can be put in relation with the cooling and heating rates of the rate equations of cooling, see [39]. In particular, for Δ < 0 then κ _+ 1^b > κ_–1^b and in this case, in the absence of the cavity coupling, the motion would be cooled.