Weak value amplification of atomic cat states

We show the utility of the weak value amplification to observe the quantum interference between two close lying atomic coherent states in a post-selected atomic cat state, produced in a system of $N$ identical two-level atoms weakly interacting with a single photon field. Through the observation of the negative parts of the Wigner distribution of the post-selected atomic cat state, we find that the post-selected atomic cat state becomes more nonclassical when the post-selected polarization state of the single photon field tends toward becoming orthogonal to its pre-selected state. We show that the small phase shift in the post-selected atomic cat state can be amplified via measuring the peak shift of its phase distribution when the post-selected state of the single photon field is nearly orthogonal to its pre-selected state. We find that the amplification factor of 15 [5] can be obtained for a sample of 10 [100] atoms. This effectively provides us with a method to discriminate two close lying states on the Bloch sphere. We discuss possible experimental implementation of the scheme, and conclude with a discussion of the Fisher information.


I. INTRODUCTION
The weak value amplification of the observables is finding increasing number of applications in the study of a variety of physical systems [1][2][3][4][5]. Although originally formulated for quantum systems, many past and current applications include applications to classical light beams. For example the first observation of the weak value amplification was in the context of a Gaussian beam propagating through a birefringent medium [6]. Other important applications of weak value measurements include observation of spin Hall effect of light [7], Goos-Hänchen shifts and various generalizations [8][9][10], angular shifts of light beams [11], enhancement of interferometric results [12]. It is intriguing that a concept formulated for quantum systems has so many profound applications in the context of classical light beams. Aiello showed in a formal way how weak value measurements work for beams of light [13]. Lundeen and coworkers used weak value measurements to get the wavefront of a single photon [14]. Steinberg [15] proposed the applications in the measurement of interaction between two Fermions. Pryde et. al. experimentally determined weak values for a single photon's polarization via a weak measurement [16]. Starling et. al. used the weak value measurements to enhance frequency shift resolution in a Sagnac interferometer [17]. Jordan et. al. examined the Heisenberg scaling with weak measurement from a quantum state discrimination point of view [18].
In this paper we show the great utility offered by weak value measurements for studying quantum mechanical cat states for atoms. The cat states are the linear superposition of two coherent states on the Bloch sphere The quantum interferences in cat state are most prominent if the two coherent states are close on the Bloch sphere [19,20]. The study of quantum interferences is greatly aided by the weak value measurements otherwise these are difficult to observe. The weak value measurements give us the capability to resolve two close lying coherent states. We look at the interaction of a single photon with an ensemble of atoms prepared in a coherent state [21,22]. The interaction produces an entangled state of the photon polarization variables with the coherent states of the atomic ensemble. We use preselection and postselection of the polarization states of the photon. The postselected polarization is nearly orthogonal to the input polarization. This enables us to magnify the weak values associated with the measurements of the phase ϕ.
The organization of this paper is as follows: In Sec. II, we introduce the model of the interacting atom-field system. In Sec. III, we make a weak measurement on the atom-field system so that the post-selected atomic cat state is generated. In Sec. IV, we present the variation of the Wigner distribution of the post-selected atomic cat state when the overlap of the initial and final states of the field changes. In Sec. V, we show that the small phase shift in the post-selected atomic cat state can be amplified by choosing nearly orthogonal pre-selection and post-selection of the single photon field. We conclude our paper in the final section.

II. ATOMIC SYSTEMS AND THE EFFECTIVE INTERACTION HAMILTONIAN
We consider an ensemble of N identical two-level atoms interacting with two orthogonally polarized modes of a single photon field with frequency ω f denoted by creation (annihilation) operators a † − , a † + (a − , a + ) as shown in Fig. 1(a). The two-level atoms have degenerate ground states |g ± and excited states |e ± , separated by an energy of ω 0 . According to the angular-momentum selection rules, the transitions |g + ↔ |e + and |g − ↔ |e − are forbidden, only the transitions |g + ↔ |e − and |g − ↔ |e + are allowed. Moreover, the levels |g + and |e − are coupled by the field mode a − , and the levels |g − and |e + are coupled by the field mode a + . Their coupling strengthes are G − and G + , respectively. The Hamiltonian of the combined system of the atoms and the field [22] takes the form In Eq. (2), the first two terms are the atomic excitation Hamiltonian, the third term is the free field Hamiltonian, the last two terms are the atom-field interaction Hamiltonian. In the dispersive limit in which the detuning between the atomic transitions and the field modes is much larger than the atom-field coupling strengthes i.e., Note that there are no couplings between the ground states |g ± i (i = 1, · · · , N ) and the excited ones. For simplicity, we assume that the coupling strengths have identical amplitudes |G − | = |G + | = G. If one starts from the initial ground state, the relevant interaction Hamiltonian between the atoms and the field modes is written as where is the field operator, is the collective atomic operator, and φ 0 = G 2 /∆ is the coupling constant between N z and J z . Another important example of optical transitions leading to the interaction Hamiltonian (4) would be three level atoms [21] with ground levels |± coupled to an excited state |e by a far off resonant field, as shown in Fig. 1(b). Under the assumption that the frequency separation between |± is such that the two photon Raman coupling between |+ and |− is negligible, the effective interaction between the atoms and the field is described by Eq. (4).
In what follows we assume that the initial state of the atomic sample is an atomic coherent state |θ, φ with a definite angular momentum value j, which can be created by rotating the ground state |j, −j on a Bloch sphere by an angle θ around an axis defined as n = (− sin φ, cos φ, 0) [32] where for which all atoms are in the ground state. The atomic coherent state also can be expressed as [32] |θ where |j, m is the simultaneous eigenstate of J 2 and J z , where j = N 2 , m = −j, −j + 1, · · · , j − 1, j, and N is a total number of atoms in the sample. The atomic coherent states are in general not orthogonal except for antipodal points. The modulus squared of the inner product of two atomic coherent states is where Θ is the angle between the directions (θ, φ) and (θ ′ , φ ′ ) on the Bloch sphere, and cos Θ = cos θ cos θ ′ + sin θ sin θ ′ cos(φ − φ ′ ). For the special cases (θ ′ − θ = ±π, φ ′ = φ; or θ ′ = θ = π/2, φ − φ ′ = ±π), cos Θ = −1, Θ = π, | θ, φ|θ ′ , φ ′ | 2 = 0, the two atomic coherent states are orthogonal. The atomic coherent states can be produced by classical driving fields of constant amplitude [33,34]. We assume that the initial state of the single photon field is given by where the coefficients c + and c − are the probability amplitudes to find the light in the states |1 + , 0 − and |0 + , 1 − , respectively, and |c + | 2 + |c − | 2 = 1. Note that the |Ψ f corresponds to an elliptically polarized light. When c + = 0 or c − = 0, the |Ψ f is circularly polarized.
When c + = c − , the light is x polarized. Note that single photons can be heralded [35,36] or a single nitrogenvacancy color center in a diamond nanocrystal [37,38]. Thus the initial state of atom-field system is |θ, φ |Ψ f , which is a product state of the two subsystems of the field and the atomic ensemble.

III. WEAK INTERACTION OF THE ATOMIC ENSEMBLE WITH THE SINGLE PHOTON FIELD AND THE POST-SELECTED ATOMIC STATE
We next study the evolution of the system under the interaction (4). This is given by where t is the atom-field interaction time. Expression (12) indicates that the atoms are entangled with the single photon field due to the atom-field coupling. The two coherent states in (12) differ in phase by 2Ω, Ω = φ 0 t. The interaction of the single photon with the medium is expected to be weak and hence Ω is small. Thus the question of distinguishing between two such coherent states arises. It is here that the weak value measurements offer great advantage. We make a preselection and postselection of the states of the polarization of the single photon. For simplicity we assume that initially the field is linearly polarized with c + = c − = 1 √ 2 i.e., |Ψ f = 1 √ 2 (|1 + , 0 − + |0 + , 1 − ). We take the postselected polarization state as in which γ is a small angle, and is controlled by the polarizer. Hence the overlap between the pre-selected and post-selected states of the field is which depends on the parameter γ. When γ = 0, Ψ ph |Ψ f = 0, the pre-selected and post-selected states of the field are orthogonal. For small γ, the pre and postselected states are nearly orthogonal. After the weak interaction between the atoms and the field, we project the state of the system |Ψ at−f onto a final state of the single photon field |Ψ ph , we obtain the normalized atomic Schrödinger cat state which is a superposition of two distinct atomic coherent states |θ, φ + Ω and |θ, φ − Ω rotating in opposite directions in phase space. Thus the weak interaction of the atoms with the single photon field, followed by the state-selective measurement on the single photon field, has transformed the initial atomic coherent state |θ, φ into a post-selected atomic cat state with two components |θ, φ + Ω and |θ, φ − Ω with small opposite phase shifts Ω. Here N is the normalization factor given by In what follows we study the post-selected state for θ = π/2 and φ = 0. This lies in the equatorial plane and has maximum coherence. In spin language this means that the spin is x polarized at t = 0.

IV. THE WIGNER FUNCTION OF THE POST-SELECTED ATOMIC CAT STATE
In this section, we use the Wigner function to quantify the nonclassicality of the post-selected atomic cat state.
The Wigner function of the atomic cat state is defined as [39] W (α, β) = 2j + 1 4π where is the state-multipole operators [40], the Wigner 3j symbol, K is an integer taking values 0, 1, 2, · · · , 2j, −K ≤ Q ≤ K, and Y KQ (α, β) is the spherical harmonic. It can be interpreted as a quasiprobability distribution in phase space. It satisfies the normalization condition After some calculations, we obtain the following result for the Wigner function (20) in which the first term and the second term correspond to the contributions of the individual coherent states | π 2 , Ω and | π 2 , −Ω in the atomic cat state, respectively, the last term represents the interference between the two atomic coherent states.
We now demonstrate the importance of the weak measurements and postselections in discriminating two co-herent states lying nearby. For illustration purpose we choose the small phase shift Ω = π/100 = 1.8 o and the total number of the atoms N = 10 (j = 5). The Wigner function for the atomic cat state on the Bloch sphere for several different post-selected values of γ is plotted in Fig. 2. We notice that the post-selected state for γ = π/2 can be hardly distinguished from the initial state. The post selection is useful here. We show the Wigner functions of several post-selected states by choosing γ values such that the pre-selected and post-selected polarization states are nearly orthogonal. For decreasing values of γ one sees more and more negative regions in the Wigner function signifying that the post-selected state becomes more and more nonclassical. The Wigner function can be determined by tomographic reconstruction [41][42][43].

V. AMPLIFICATION OF THE PHASE SHIFT OF THE POST-SELECTED ATOMIC CAT STATE
In this section, we show that how the small phase shift Ω in the post-selected atomic cat state induced by the weak interaction of the atomic sample with the single photon field can be measured by studying the phase distribution of the atomic cat state.
The phase distribution P (ϕ) for the atomic cat state [44], i.e., the probability distribution P (ϕ) of finding the The probability distribution P (ϕ) as a function of ϕ for different γ when Ω = π/100 and j = 5. The purple long-dashed, green dotdashed, red dotted, blue short-dashed, black solid curves correspond to γ = π/2, π/30, π/60, π/100, 0, respectively. system in the state | π 2 , ϕ , is given by The figure 3 shows that the phase distribution P (ϕ) as a function of ϕ for different values of γ for Ω = π/100 and j = 5. As the value of γ is decreased from π/2 to 0, the phase distribution P (ϕ) gradually changes from a single peak at ϕ = 0 to two symmetric peaks with a dip at ϕ = 0, which is due to the destructive interference between two weakly separated coherent states in the atomic cat state. The scaled left peak shift |shift/Ω| from ϕ = 0 as a function of γ is shown in Fig. 4. It is noted that the scaled peak shift |shift/Ω| increases with decreasing γ. When γ = π/100, |shift/Ω| ≃ 15, the amplification factor for the weak value Ω is about 15. If there is 100 atoms in the atomic sample (j = 50), the phase distribution P (ϕ) as a function of ϕ for different values of γ is shown in Fig. 5. The scaled left peak shift |shift/Ω| from ϕ = 0 as a function of γ is shown in Fig. 4. As the value of γ is decreased, the scaled left peak shift |shift/Ω| increases. When γ = π/100, |shift/Ω| ≃ 5.8, the amplification factor for the weak value Ω is about 5.8. Note that the amplification factor for the weak value Ω decreases with increasing the total number N of the atoms. This is because the modulus squared of the overlap of the two atomic coherent states | π 2 , Ω and | π 2 , −Ω decreases when the total number N of the atoms becomes larger, which is shown below. Using Eq. (10), one gets | π 2 , Ω| π 2 , −Ω | 2 = cos 4j Ω, FIG. 5: (Color online) The probability distribution P (ϕ) as a function of ϕ for different γ when Ω = π/100 and j = 50. The purple long-dashed, green dotdashed, red dotted, blue short-dashed, black solid curves correspond to γ = π/2, π/30, π/60, π/100, 0, respectively.

VI. CONCLUSIONS
We have demonstrated how the nonclassical interference between two slightly separated atomic coherent states in a post-selected atomic cat state can be observed via making a weak value measurement on a system containing N two-level atoms weakly coupled to a single photon field. We find that the negative region of the Wigner distribution of the atomic cat state is increased when the pre-selected and post-selected states of the single photon field are closer to orthogonal. We show that the weak value measurements can lead to peak shift in the phase distribution of the atomic cat state that can be 15 times [for 10 atoms] the phase shift in the atomic cat state when the initial and final states of the signal photon field are nearly orthogonal.