1. Introduction

The most striking features of quantum physics, used as resources of modern quantum technologies [1–4], are those that can not be explained by any classical theory. These quantum states and quantum operations are naturally called nonclassical. Nonclassical quantum states of linear harmonic oscillators are best recognized by the presence of negative or singular values in their semiclassical quasi probability distributions [5–7]. An important kind of nonclassicality is bound to negative values of one particular quasidistribution, the Wigner function, and is considered a fundamental resource of advanced quantum information protocols, such as quantum computing [8–10]. The presence of negativity in Wigner function is directly measurable and it ensures that the state can not be replaced by a mixture of Gaussian quantum states.

Nonclassical processes are those that transform classical states into nonclassical ones. Again, the important nonclassical processes are those capable of generating states with negativity in Wigner function. Such processes offer a qualitative difference over both classical and linear dynamics of the oscillator and are necessary for preparation and processing of states for quantum information processing. The simplest example of such highly nonclassical process is a conditional addition (creation) of single quantum [11–15]. For classical states, this process always generates negativity in the Wigner function, even when the system was initially in a thermal state with arbitrary energy [5, 6]. This physical process is solely responsible for generation of optical single photon states that are the basic building blocks of all the optical discrete variable quantum information applications [16]. A conceptually similar process is a conditional subtraction (annihilation) of a single quantum [17, 18]. This process can be also used for generation of nonclassical states with non-positive Wigner function but for that that the initial state of the system already needs to be in a nonclassical squeezed state. As such, the conditional subtraction only converts one kind of nonclassicality into another one and it can not create nonclassicality in classical states on its own.

However, it is also possible to have a process that appears classical, but ultimately is not. Absorption is an elementary physical process in which energy of a given system S (for example, an oscillator) is collected by another system A (for example, a two-level system). From a macroscopic perspective, the rate of absorption does not depend on the energy of system S, which is damped linearly. This linear absorption is a classical process, because classical states of oscillator S, such as thermal states, can not be converted into non-classical ones. However, this model can be considered an approximation, in which the life-time of the excited level of the two level system is negligible. We can abandon this approximation and consider a situation, in which the absorption is performed by a single two-level system one quantum at a time. Such process is intrinsic to several modern experimental platforms, including for example single trapped ions which can absorb their mechanical energy [2], a single two-level atoms interacting with an electromagnetic radiation in optical resonator [1], or a single flux qubits absorbing electric energy of super-conducting LC circuit [4]. In the complete quantum picture, all these interactions are described within rotating wave approximation by the basic Jaynes-Cummings (J-C) model [19, 20]. Its presence was evidenced by the early experiments studying Rabi oscillations of two-level system without any measurements of the oscillator [21–23].

In this paper we study the state of the harmonic oscillator, initially in a thermal equilibrium, that is coupled to a single two-level system via Jaynes-Cummings interaction. In contrast to previous works and experiments, we consider only measurements performed on the oscillator, disregarding the two-level system entirely, see Fig. 1. We show that this deterministic dissipative process is highly nonclassical, capable of generating states with strongly negative Wigner functions even when the oscillator was initially found in a classical thermal state. This is the first account of nonclassicality of this kind being generated by a deterministic process without an external source of coherent energy. This is very important for harmonic oscillator quantum systems which are difficult to prepare in the ground state, such as vibrational modes of trapped molecular ions, solid state systems, and superconducting circuits. The noisy nature of those oscillators makes observation of any non-classical effect very difficult and the method we propose in this paper may be one of the first opportunities to do so. It should be stressed that it is not that the process is robust against the noise in the initial thermal state, but that it is actually driven by it. This is witnessed by the need for some minimal amount of thermal fluctuations before the nonclassicality can be observed. All these unexplored aspects of the absorbtion by a two-level system cannot be obtained in single-photon subtraction experiments [3]. The proposed process of generation of nonclassicality is also robust to an imperfect initial population of the two-level system, to its dephasing and to the detuning of the coupling laser field. We propose two experimental platforms for feasible tests of this effect. Repeated applications of this process can open a way to a qualitatively new way of generating non-classical states of quantum oscillator suitable for quantum technology.

 

Fig. 1 Absorbtion of a quanta from quantum oscillator by a two-level with thermal populations unconditionally generates negative Wigner function from initially thermal state of the oscillator.

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2. Nonclassicality from absorption

In quantum physics, the elementary absorbtion of energy of the oscillator by a single two-level system is an unitary process described by a full Jayness-Cummings Hamiltonian H = H₀ + H_I, where H₀ is the free evolution Hamiltonian of the joint system and H_I describes the interaction between it parts [19–21]. The free evolution Hamiltonian $H_0=\frac{\overline{h}\omega {\sigma }_Z}{2}+\overline{h}\nu a^{†}a$ depends on transition frequency $\omega =\frac{E_e-E_g}{\overline{h}}$ of the two-level system, where E_g and E_e are the energies of the ground |g〉 and the excited |e〉 states, respectively and ν denotes the frequency of the oscillator. The operator σ_Z is inversion operator for the two-level system and a^†, a are oscillator’s creation and annihilation operators of the oscillator. We denote Δ = ω − ν as the detuning parameter. In the rotating-wave approximation, Jaynes-Cummings interaction Hamiltonian H_I = ih̄g(σ₊a − σ₋a^†) describes the processes of energy exchange between the oscillator and the two-level system, where g is interaction strength and σ₊, σ₋ are two-level raising and lowering operators, respectively.

For the resonant absorbtion (Δ = 0) the interaction between the two systems can be represented by unitary operator

For the initial thermal state of the oscillator given by density matrix ${\rho }_{\mathit{TH}}={\sum }_{n=0}^{\infty }{\rho }_{\mathit{TH}}(n)|n〉〈n|$, where ${\rho }_{\mathit{TH}}(n)=\frac{{\overline{n}}^n}{{\left(1+\overline{n}\right)}^{1+n}}$ is Bose-Einstein statistics, the absorption event unconditionally results in density matrix

As an indicator of higher nonclassicality of the oscillator state ρ(t), we use negative values of Wigner function. The presence of negative values ensures that the state is incompatible with a mixture of Gaussian states, which can be prepared only with quadratic nonlinearities. The negativity is also a necessary aspect of quantum states needed for advanced quantum information tasks [8–10]. Wigner functions can be obtained as the mean value W(x, p; t) = Trρ(t)D(x, p)ΠD^†(x, p), where $\mathrm{\Pi }=\frac{2}{\pi }{\sum }_{n=0}^{\infty }{\left(-1\right)}^n|n〉〈n|$ is the parity operator [25], and x and p are the respective values of quadrature operators X and P, with [X, P] = 2i, and D(x, p) = exp[−i(xP + pX)] is the displacement operator. When we look just at the point of origin the formula can be explicitly evaluated and simplified to

Negativity of Wigner function is a well established sufficient criterion of nonclassicality. It is also a necessary prerequisite for some crucial quantum information tasks [8,9]. For Fock states |n〉 of the oscillator, the Wigner function $W_n(x,p)=\frac{2}{\pi }{\left(-1\right)}^n{L}_n(4x^2+4p^2)\text{exp}(-2x^2-2p^2)$ exhibits negative values. Here L_n corresponds to the Laguerre polynomial Here L_n corresponds to the Laguerre polynomial of n-th order. As the Fock number n increases, so does the number of negative regions and their distance from the point of origin. These regions of negativity are a definite sign of nonclassicality, meaning that the corresponding Wigner function can not be interpreted as a probability density of some hidden variables, and the state can not be obtained as a statistical mixture of Gaussian states.

When the oscillator in a thermal state is subject to the nonlinear absorption (5), the resulting state becomes strongly non-classical, which can be witnessed by observing negative values at the point of origin, see Fig. 2(a). These values arise unconditionally - after the interaction, the two level system is discarded without any measurement. The negativity is therefore not a direct consequence of correlations between the systems, but it appears purely as the effect of nonlinear dynamics incompatible with the classical linear absorption processes. The negative values appear as irregular oscillations and their presence is mostly independent on the mean energy of the initial thermal state with one exception: the energy can not be too low, the mean number of excitations roughly needs to surpass n̄ ∼ 2. The resulting nonclassicality is therefore clearly fuelled by the initial thermal energy. The minimum energy requirement can be easily understood by considering the properties of W(0, 0; t). This value is negative if the photon number distribution of the generated state has higher total weight of odd components compared to even ones. Since the vacuum term 〈0|ρ(t)|0〉 can never be reduced, as long as the initial state has 〈0|ρ(0)|0〉 ≥ 1/2 the produced state can never be nonclassical. This means that a necessary condition to observe any nonclassicality is initial state with n̄ > 1. In practice this requirement is slightly stricter because of higher Fock numbers and their more involved dynamics, but the basic reasoning remains. The reason why the nonclassicality is otherwise mostly independent on the energy of the oscillator may be gleaned from (6). There we can see that the initial mean energy serves only as a backdrop on which the nonclassical oscillating pattern produced by the absorption performs. As n̄ increases, the term n̄ⁿ/(1 + n̄)¹⁺ⁿ becomes more and more uniform until it ends up completely irrelevant.

 

Fig. 2 Negative values of W(0,0) for initial thermal states with mean energy n̄ undergoing absorption of a single photon with accumulated parameter gt. (a): ideal single atom absorption with the two level system in a ground state. (b): ideal single atom absorption with the two-level system being in a thermally excited state with P_e = 0.3. (c): single atom absorption in a detuned regime with g = 1, Δ = 5 and the two level system in a ground state. (d): single atom absorption with the two level system initially in a ground state undergoing loss (spontaneous emission into other modes) with g = 1 and γ = 0.015g.

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The negative values also appear only for stronger absorption corresponding to gt ∼ π. The negative values of Wigner function thus cannot be observed in a short-time limit of the Jaynes-Cummings interaction. The process has both thermal and interaction length observability thresholds, which is the consequence of modulation of the Bose-Einstein statistic by the nonlinear terms. Maximum negativity for a given n̄ can be found by optimization of gt, which needs to be performed numerically.

The appearance of negative values does not require the two-level system starting in the ground state. As demonstrated in Fig. 2(b)., the negative values of Wigner function are generated even when the absorbing two-level system is initially in a state with relatively high energy. In this case the the two level system was initially in mixed state P_e|e〉〈e| + (1 − P_e)|g〉〈g| and the absorption transformed the state of the oscillator into

The generation of negativity is a resonant effect and it therefore depends on detuning and dephasing of the two level system. To analyze this features we need to consider the total Hamiltonian in the rotating wave approximation H = h̄Δσ_z + ih̄g(σ₊a − σ₋a^†) and find the total dynamics of the system by solving master equation

The nonclassicality of the state produced by the single atom absorption is dependent on the initial thermal fluctuations. There is a certain minimal energy threshold that needs to be surpassed before the absorption produces a state with negative Wigner function. Then, as the thermal energy increases, so does the number of regions with negative values; see Fig. 3. Note that since the resulting states are phase insensitive, these figures provide the complete information about the produced quantum states. The states are strongly mixed. The mixedness, however, does not eliminate the highly nonclassical effects. The number of negative regions for a given gt is not necessarily an indicator of strength of the nonclassicality. On the other hand, the set of gt values which produce negativity gets enlarged with increasing n̄, which indicates that for higher initial thermal energy the highly nonclassical states are more common. In this sense it is safe to say that the produced nonclassicality is powered by the initial thermal fluctuations.

 

Fig. 3 Negative values of W(x, 0) undergoing the ideal single photon absorption with accumulated parameter gt. The two level system initially in a ground state. (a): n̄ = 2. (b): n̄ = 5. (c): n̄ = 10. (d): n̄ = 20.

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For each particular initial energy, the value of Wigner function at the origin oscillates between positive and negative values as the interaction progresses. This is the consequence of various lengths of the interaction differently accenting either odd or even Fock states. Fig. 4 demonstrates this effect by showing Wigner functions and photon number distributions P(n, t) = 〈n|ρ(t)|n〉 for the thermal state with n̄ = 10 and two different interaction lengths gt = 5.5 and gt = 2π. In this example, the odd Fock states are dominant and even Fock states are suppressed for ρ(5.5), while the reverse holds true for ρ(2π). We can see in Fig. 4(a) that the negativity is fairly strong, being of the same order as the maximal positive values of the Wigner function.

 

Fig. 4 Thermal states undergoing the ideal single photon absorption with the two-level system being initially in a ground state. (a) W(x, p) for n̄ = 10 and gt = 5.5. (b) W(x, p) for n̄ = 10 and gt = 2π. (c) P(n) for n̄ = 10 and gt = 5.5. (d) P(n) for n̄ = 10 and gt = 2π. The line in figures (c) and (d) follows the coefficients of the initial thermal state.

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Let us now discuss in detail the differences between nonclassicality from saturable absorption and nonclassicality traditionally produced in quantum optical experiments by either photon subtraction [17, 18] or photon addition [11–15]. The main drawback of photon subtraction is that it needs the initial states to be nonclassical, specifically squeezed, in order to produce states with negative Wigner function. Subtracting a photon from a thermal state produces only a classical state. Photon addition is more nonclassical in this regard as it can produce negative Wigner functions even from thermal states. The nonclassicality is caused mainly by the artificial removal of the vacuum term. This, in combination with monotonously increasing the weight of higher Fock components, leads to overall increase of energy and reduction of purity of the state. The saturable absorption on the other hand generates nonclassicality by modulating the weights of Fock coefficients, reducing the state’s energy and increasing its purity. Another important difference lies in the probability of success of the operation. Single photon subtraction and addition experiments operate with a very low success probability of about 10⁻⁴ [13] whereas the saturable absorption is a deterministic process.

3. Repeated absorptions

The process of saturable absorption can produce quantum states that are nonclassical. Let us now investigate what specific states can be created in this way. If we apply the absorption repetitively, using a separate two-level system for each event, the initial thermal state will slowly converge towards vacuum state, unless the interaction constant is set to $gt=\pi \sqrt{k}$ where k is a positive integer. In that case the Kraus operator for absorption, A_eg (3), yields zero when applied to certain Fock states and, irrespectively to the initial thermal energy, the state is driven towards the form

Fig. 5 shows the evolution of the initial thermal state subjected to the sequence of absorption events. We can see that the Fock coefficients not corresponding to squares of integers (for k = 1) or their doubles (for $k=\sqrt{2}$) quickly vanish. This can be exploited for practical purposes, such as state preparation. For example, using sequences of absorption events with different interaction constants and different limit distribution can be used to depopulate all but Fock states, thus allowing generation of an impure number state p′_N+1|N + 1〉〈N + 1|+ p′₀|0〉〈0|. The main limitation is that the contribution of the vacuum term p₀ can never be lowered by the deterministic method. On the other hand, it can be completely removed by probabilistically subtracting a single quantum from the oscillator. In optics, this can be effectively done even with detectors with a low quantum efficiency. This allows preparation of an arbitrary number state |N〉 from a thermal state by a sequence of deterministic absorption events and only a single feasible probabilistic measurement. Such state can be then used as a universal resource for quantum information processing [28]. It should be noted that this kind of preparation of a specific state is largely independent on the choice of the initial thermal state. All thermal states lead to the same output state and the only difference is the success probability of the last conditional subtraction.

 

Fig. 5 The number distribution P(n) of a thermal state with n̄ = 5 undergoing M absorption events with coefficient gt = π.

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4. Nonclassicality from emission

A complementary process closely related to the considered process of absorption is associated with emission of a single quantum. A complementary effect appears when the two-level system is initially in the excited state. There are several ways of possible implementation. One is having the two-level system initially in excited state and using the standard Jaynes-Cummings interaction. Another way keeps the two level system in the ground state, but applies the counter-rotating J.-C. Hamiltonian H_I = h̄κ(a^†σ₊ + aσ₋). Physically, this corresponds for example to driving the first blue-detuned sideband instead of the red one for a trapped ion. In this case, the excitation of the two level system is accompanied by emission of a quanta, rather than its absorption, and the nonlinear dynamics is qualitatively different.

As the oscillator receives the single quantum of energy, the state ρ_th is evolved to:

 

Fig. 6 Negative values of W(x, 0) for thermal states undergoing the evolution in the blue-detuned sideband regime. (a): n̄ = 2. (b): n̄ = 5. (c): n̄ = 10. (d): n̄ = 20.

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5. Experimental feasibility

The verification of both the fundamental aspects of nonclassical saturable absorption and the ability to prepare new highly nonclassical states is an interesting prospect. Currently, the experiment can be performed with cavity and circuit quantum electrodynamics or with mechanical states of trapped ions. For a single ion trapped in a Paul trap, coupling between the ion’s electronic states on a long-lived transition and ion’s motion in the Lamb-Dicke regime naturally appears in the form of J-C Hamiltonian. Presented calculations suggest that desired Wigner distributions with multiple negativity regions would be observable with experimental parameters and phonon numbers corresponding well to the Lamb-Dicke approximation [2]. Considering typical trapping parameters corresponding to ω_trap ∼ (2π) 1MHz, Lamb-Dicke parameter η ∼ 0.1, carrier transition Rabi frequency Ω ∼ (2π) 100kHz and laser-ion interaction on long-lived transition with coherence times at the order of milliseconds [29], coupling parameter g on the first motional sidebands would correspond to g ∼ ηΩ = (2π) 10kHz. The number of coherent Rabi cycles for a single ion in such setup can easily reach gt > 10 without any substantial decrease of the oscillation contrast [2] and thus allows the proposed observation of several Wigner function negativity areas, as depicted in Fig. 3. Furthermore, the motional state of trapped ion after laser cooling conveniently corresponds to thermal distribution as a result of thermal equilibrium with an external reservoir [2]. The resulting Wigner distribution can be reconstructed using one of the well developed methods based on application of controlled coherent displacements to ion’s motion followed by the measurement of its populations in the Fock state basis [23]. Recent experimental results suggest that motional populations of up to n=10 phonons can be reliably estimated [30]. Together with an unprecedent control and measurement sensitivity to individual recoils this further demonstrates the convenience of ion trapping setups for experimental tests of the proposed generation of nonclassical states [31].

Alternatively, the proposed scheme could be tested in any other experimental platform where prominent interaction is governed by the Jaynes-Cummings Hamiltonian and which provides necessary degree of control over this coupling. In cavity-QED setups utilizing high-Q microwave resonator, for example in [32–34], a high quality of direct probing of Wigner function will also allow to observe highly nonclassical effects of the absorption process. Thermal state of microwave field inside the cavity can be easily obtained from an external thermal source weakly coupled to the cavity. Recently, a preparation of high-quality quantum states with multiple negative islands of the Wigner function appeared in circuit-QED experiments, where well controllable J-C interaction is also available [35–39].

6. Conclusion and outlook

We have identified a highly nonclassical process capable of transforming highly classical states into highly nonclassical ones. The process, the naturally occurring saturable absorption by a single two-level systems, is set apart from the previously known nonclassical processes (single quantum addition, for example) by being deterministic and requiring no external coherent sources of energy. Both of these properties contribute towards its general feasibility. The proposed scheme can be easily implemented in most of existing systems governed by the fundamental J.-C. interaction and becomes extremely useful for platforms where reaching the ground state of the oscillator’s mode corresponds to a challenging task. Beyond the fundamentally interesting discovery of a deterministic highly nonclassical process, the presented work could be further extended by considering a sequence or combination of the absorption and emission events to further improve quality of the generated states. The presented approach will likely pave the way for further studies of other available nonlinear processes and their tests for analogical, or even more interesting, thermally induced nonclassical effects. We foresee a direct applications of our scheme for generation and tests of macroscopic quantum nonclassicality and for preparation of nonclassical states suitable for quantum information processing [9, 28] or precision measurements [40].