A photon counting and a squeezing measurement method by the exact absorption and dispersion spectrum of Λ-type Atoms

Recently, the master equations for the interaction of two-mode photons with a three-level Λ-type atom are exactly solved for the coherence terms. In this paper the exact absorption spectrum is applied for the presentation of a non-demolition photon counting method, for a few number of coupling photons, and its benefits are discussed. The exact scheme is also applied where the coupling photons are squeezed and the photon counting method is also developed for the measurement of the squeezing parameter of the coupling photons.

the quantized fields in opto-cavity mechanics is another example for the full-quantum approach which is studied by Huang and Agarwal (2011). The destructive detection of photons has been investigated theoretically and experimentally. But non-demolition detection of photons (Braginsky and Khalili 1996) has until now been an interesting ultimate goal of some optical measurement methods (Grangier et al. 1998). In 2012, Serge Haroche and coworkers have been shown (Sayrin et al. 2012) that interaction of microwave photons, trapped in a superconducting cavity, with Rydberg atoms crossing the cavity, illustrates a non-demolition photon counting. In 2013, Andreas Raiser (Raiser et al. 2013) presents another method for non-demolition detection of photons which are passing through a superconductive cavity resonator that includes rubidium atoms. Haroche et al. (Sayrin et al. 2012) and Naeimi et al. (2013) investigated a photon counting and squeezing parameter measurement (for photons trapped in a quantum cavity) by measure the properties of a beam of atoms interacted with an array of cavities. But photon counting by measure the properties of another photons (or field) which are passing through the cavity, have never been investigated to our best of knowledge. In this paper, we present an exact analytical non-demolition photon counting method (for photons inside a cavity) by investigating the absorption profile of probe field. A full-quantum model of EIT is investigated for an ensemble of Λ-type three-level atoms, in which the probe and coupling fields are quantize. Interaction of a Λ-type three-level atom with the quantized electromagnetic fields is investigated using the Jaynes-Cummings model (Khademi et al. 2015). The Jaynes-Cummings interaction Hamiltonian is applied for each of the coupled levels. In this case, the exact master equations are investigated and solved in a steady-state without any WFA (Khademi et al. 2015). An exact form of absorption and dispersion spectra are obtained for the probe fields which are not generally weaker than the coupling field. It is shown that the EIT obtained for the probe fields is either weaker or stronger than that of the coupling field.
Moreover, profile of the absorption and dispersion spectra are shown to depend on the number of coupling photons so that the number of coupling photons could be measured using the absorption spectrum of the probe photons. This scheme is applied for the presentation of a non-demolition photon counting method. The present method is applied to the squeezed coupling photons. Straightforwardly, it is shown that the exact absorption and dispersion spectra drastically depended on squeezing parameter of the coupling photons. This scheme is also applied for presenting measurement of the squeezing parameter.
In "A review on the exact model" section, a review on the exact model of the fullquantum interaction of quantized electromagnetic fields with a Λ-type three-level atom will be presented. More details are found in reference (Khademi et al. 2015). The master equations in the steady-state, their exact solutions, a schematic experimental setup and notations are also introduced. "Photon counting by an ensemble of Λ-type three-level atom" section is devoted to a photon counting method in terms of the measurement of absorption spectrum. In "Measuring squeezing of trapped coupling photons" section, the exact probe coherence term is obtained where the coupling photons are squeezed. It is shown that the squeezing parameter is also measurable by the measurement of absorption and dispersion spectrum. The last section is devoted to the "Conclusions".

A review on the exact model
In this section a review on the exact model of a three-level Λ-type atom interacting with two quantized electromagnetic field (Khademi et al. 2015) is presented. The master equations, notations, experimental setup and some solutions and results are used in the next sections.
Suppose that, in cavity quantum electrodynamics, the quantized probe and coupling fields (photons) interact with a three-level Λ-type atom (see Fig. 1a). The interaction Hamiltonian of this system in the interaction picture is given by: where g 1 = ℘ ab ·ε 1 E 1 / and g 2 = ℘ acε2 E 2 / are interaction strength of the probe and coupling fields, respectively, and E i = ( ν i /2ε 0 v) 1/2 . In this case, v is cavity volume and ℘ ab = e�a|r b and ℘ ac = e�a|r|c� are matrix elements of atomic dipole moments, induced by the electromagnetic fields. â 1 â † 1 and â 2 â † 2 are annihilation (creation) operators for the probe and coupling photons, respectively. σ ij = |i� j is atomic transition operator from j → |i�. In Eq. (1), � 1 = ω ab − ν 1 (� 2 = ω ac − ν 2 ) is detuning between the frequency of probe (coupling) and the atomic transition frequency |a� → b (|a� → |c�).
Assume the system is initially in the ground state b and the electromagnetic fields for the probe and coupling fields are in the states |n 1 � and |n 2 �, respectively. Therefore, the initial state of total system is given by b, n 1 , n 2 . After an atom-field interaction, one photon with frequency of ν 1 is absorbed and the atom is then transited into the higher level |a� and state of the total system changes to |a, n 1 − 1, n 2 �. Due to the spontaneous or induced emission, the atom in the state |a� is transited into another level |c� and one photon with frequency of ν 2 is emitted and state of the total system changes to |c, n 1 − 1, n 2 + 1�. The master equations are obtained as: (1) V = − g 1 σ ab a 1 e i� 1 t + a † 1 σ ba e −i� 1 t − g 2 σ ac a 2 e i� 2 t + a † 2 σ ca e −i� 2 t , (2) ρ aa = −(γ 1 + γ 2 )ρ aa + ig 1 √ n 1 (ρ ba −ρ ab ) + ig 2 n 2 + 1(ρ ca −ρ ac )
An ensemble of cold three-level atoms is prepared by an optical pumping initially in the state b . The quantum cavity is filled with the three-level cold atoms as well as the n 2 number of coupling photons. The coupling photons are strongly coupled with the quantum cavity electrodynamics. The probe photons are individually injected into the cavity and absorptive atoms. Absorption of the probe photons is controlled by the number of coupling photons n 2 and measured by the detector D1. This experiment would be frequently performed for a specific number of the coupling photons trapped in the cavity. Absorption spectrums for different numbers of coupling photons are plotted in Fig. 2b, d.
The master Eqs.

Photon counting by an ensemble of Λ-type three-level atom
In this section, a non-demolition photon counting method is presented for measuring the number of coupling photons which are trapped in a quantum cavity and interact with an ensemble of three-level Λ-type atoms.
It is worthwhile to estimate the number of probe photons which is required to determine the probe absorption peak. As an example, the cavity field decay rate can be estimated as κ = 5π MHz (Raiser et al. 2013) and the coupling photons inside a high Q-factor cavity will not decay as soon as 0.1 μs. A traveling time for a probe photon passing through a cavity with dimensions about a few millimeters is also about 3 ps. Approximately, 2 × 10 4 of probe photons are passing through the cavity meanwhile the coupling photons are trapped. This number of photons is sufficient to have a good precision to determine the absorption and dispersion curves in different detuning.
It is clear in Eqs. (8)-(17) and Fig. 2 that the profile and the probe DAP in the absorption spectrum depend on the number of coupling photons. The derivative of the imaginary part of the coherence term, Eq. (17), with respect to 1 can be set to zero: to obtain the DAP δ 1 (n 2 ) = � 1Max.Abs. . DAPs are nonlinearly increased by the number of coupling photons n 2 , as plotted in Fig. 3a-c for large and small numbers of coupling photons. Figure 3a presents the relation between measurable DAPs and the large number of coupling photons.
Although Fig. 3a shows the same behaviour in the exact and WFA methods for a large number of coupling photons, there is a fine difference which is shown in Fig. 3b. But, Fig. 3c shows the difference between WFA and exact methods for the small number of coupling photons. In this condition, the exact method has more benefits than the WFA methods.
A considerable difference between the plots in Fig. 3b, c indicates that the exact method in the full-quantum model provides more correct photon numbers, even for a few number of coupling photons. Furthermore, measurement of absorption spectrum versus detuning of the probe field is a simpler method compared with other photon counting schemes. It is also a non-demolition method for weak probe fields.
To measure the small number of photons or squeezing parameters, the accuracy of DAP and DDP measurements, according to the data in Table 1, should be about 0.2 g. For the range of atomic transition frequencies, 10 MHz < g < 1 THz the accuracy should be at least 2 MHz which is larger than the new electro-optical modulator resolutions (about 1 MHz) (Veisi et al. 2015).

Conclusions
In this paper, the master equations of Λ-type three-level atom interacting with twomode quantized electromagnetic field and its exact coherence term are applied to obtain the squeezed and non-squeezed coupling photons. The following results were obtained: (1) The method was applied for presenting a photon counting and a squeezing measurement method by measuring the absorption spectrum of the probe photons. (2) The difference between the exact and WFA photon counting methods and benefits of the exact method (especially for the weak coupling photons) were demonstrated. This sensitivity increased for the larger number of coupling photons by increasing the squeezing parameter.