Reconstruction of quasiprobability distribution functions of the cavity field considering field and atomic decays

Optics Communications 400 (2017) 69–73

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Invited paper

Reconstruction of quasiprobability distribution functions of the cavity field considering field and atomic decays N. Yazdanpanah a,b,c , M.K. Tavassoly a,b,c , R. Juárez-Amaro d,e , H.M. Moya-Cessa e, * a b c d e

Atomic and Molecular Group, Faculty of Physics, Yazd University, Yazd, Iran The Laboratory of Quantum Information Processing, Yazd University, Yazd, Iran Photonic Research Group, Engineering Research Center, Yazd University, Yazd, Iran Universidad Tecnológica de la Mixteca, Apdo. Postal 71, Huajuapan de León, Oax., 69000, Mexico Instituto Nacional de Astrofísica, Óptica y Electrónica Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico

a r t i c l e

i n f o

Keywords: Atom–field interaction Measurement Master equation Quasiprobability functions

a b s t r a c t We study the possibility of reconstructing the quantum state of light in a cavity subject to dissipation. We pass atoms, also subject to decay, through the cavity and surprisingly show that both decays allow the measurement of 𝑠-parametrized quasiprobability distributions. In fact, if we consider only atomic decay, we show that the Wigner function may be reconstructed. Because these distributions contain whole information of the initial field state, it is possible to recover information after both atomic and field decays occur. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The measurement of a quantum state is a central topic in quantum optics and related fields [1,2]. Several techniques have been developed in order to achieve such goal, for instance the direct sampling of the density matrix of a signal mode in multiport optical homodyne tomography [3], tomographic reconstruction by unbalanced homodyning [4], cascaded homodyning [5], and reconstruction via photocounting [6], to cite some. There exist some proposals to measure electromagnetic fields inside cavities [7–9]. More recently Swingle and Kim [10] have shown how to reconstruct quantum states from local data and Rundle et al. [11] have shown how to reconstruct the wavefunction of any quantum mechanical system by tomographic techniques. State reconstruction in cavities is usually achieved through a finite set of selective measurements of atomic states [7] making possible to reconstruct different quasiprobability distribution functions [12]. A very basic possibility of measuring the state of quantized electromagnetic field in a cavity is by interacting it with a two-level atom and measuring the atomic inversion as the atom exits the cavity, as the field will imprint its features in such observable, for instance, if a squeezed state is initially considered, there exist so called ringing revivals of the Rabi oscillations that indicate such non-classical field [13,14]. In real experiments it is expected that dissipative processes, that have destructive effects on quantum coherences and therefore in the process * Corresponding author.

E-mail address: [email protected] (H.M. Moya-Cessa). http://dx.doi.org/10.1016/j.optcom.2017.05.001 Received 23 January 2017; Received in revised form 26 April 2017; Accepted 2 May 2017 Available online 15 May 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.

of reconstruction, may occur. Schemes involving physical processes that allow the storage of information about quantum coherences of the initial state in the diagonal elements of the density matrix have been already proposed to take care of dissipative processes [8,15]. Leonhardt and Paul [16] have shown a relation between losses and 𝑠-parametrized quasiprobability distributions and problems with the reconstruction of Wigner function have also been analyzed in Ref. [17]. Methods to reconstruct quasiprobability distribution functions are usually based on quantum state tomography [18], or on the expression [19–21] 𝐹 (𝛼, 𝑠) =

∞ ( ) ∑ 𝑠+1 𝑘 1 ⟨𝛼, 𝑘|𝜌|𝛼, ̂ 𝑘⟩, 𝜋(1 − 𝑠) 𝑘=0 𝑠 − 1

(1)

with 𝑠 as the parameter of quasiprobability function that indicates the relevant distribution (𝑠 = −1 Husimi [22], 𝑠 = 0 Wigner [23] and 𝑠 = 1 Glauber–Sudarshan [24,25] distribution functions), the states |𝛼, 𝑘⟩ are the so-called displaced number states [26] and 𝜌̂ is the density matrix. The Wigner functions of quantized motion of an ion and the quantized cavity field, were measured by Leibfried et al. [27] and Bertet et al. [28], respectively, by using the above expression. It is possible to reconstruct a quasiprobability distribution function from the above equation since there exists a direct recipe. In order to show it, let us ̂ write Eq. (1) by writing the displaced number state as |𝛼, 𝑘⟩ = 𝐷(𝛼)|𝑘⟩,

N. Yazdanpanah et al.

Optics Communications 400 (2017) 69–73

̂ where 𝐷(𝛼) is the Glauber displacement operator [24] 𝐹 (𝛼, 𝑠) =

1 𝜋(1 − 𝑠)

∞ ( ) ∑ 𝑠+1 𝑘 𝑘=0

𝑠−1

̂ ⟨𝑘|𝐷̂ † (𝛼)𝜌̂𝐷(𝛼)|𝑘⟩.

where it is defined 𝑅̂ 𝜌̂ = −𝑖𝜒 𝜎̂ 𝑧 𝑛̂ 𝜌̂ + 𝑖𝜒 𝜌̂𝜎̂ 𝑧 𝑛. ̂

(2)

(11)

One can note that

Now, in order to obtain a quasiprobability distribution function we need to displace the system by an amplitude 𝛼 and then measure the diagonal elements of (displaced) density matrix. We now aim to study the problem of reconstruction of the cavity field as studied in [29], however, not only cavity decay but also atomic decay is allowed [30]. We then want to show that it is still possible to recover whole information about the initial state through the reconstruction of 𝑠-parametrized quasiprobability distributions.

′ ′ ̂ ̂ ̂ ̂ ̂ ̂ 𝑒−(𝑅+𝐿𝐴 +𝐿𝐹 )𝑡 (𝐽̂𝐴 + 𝐽̂𝐹 )𝑒(𝑅+𝐿𝐴 +𝐿𝐹 )𝑡 𝜌̂ ( ) ̂ ̂ = 𝐽̂𝐴 𝑒−(𝑅𝐹 +2𝛾)𝑡 + 𝐽̂𝐹 𝑒−(𝑅𝐴 +2𝜅)𝑡 𝜌, ̂

in which the commutation relations [𝐿̂ 𝐴 , 𝐽̂𝐴 ]𝜌(𝑡) ̂ = 2𝛾 𝐽̂𝐴 𝜌(𝑡), ̂

[𝐿̂ 𝐹 , 𝐽̂𝐹 ]𝜌(𝑡) ̂ = 2𝜅 𝐽̂𝐹 𝜌(𝑡), ̂

̂ 𝐽̂𝐴 ]𝜌(𝑡) [𝑅, ̂ = 𝐽̂𝐴 (𝑅̂ 𝐹 𝜌(𝑡)), ̂

̂ 𝐽̂𝐹 ]𝜌(𝑡) [𝑅, ̂ = 𝐽̂𝐹 (𝑅̂ 𝐴 𝜌(𝑡)), ̂

𝑅̂ 𝐹 𝜌̂ = 2𝑖𝜒 𝑛̂ 𝜌̂ − 2𝑖𝜒 𝜌̂𝑛, ̂

𝑅̂ 𝐴 𝜌̂ = 𝑖𝜒 𝜎̂ 𝑧 𝜌̂ − 𝑖𝜒 𝜌̂𝜎̂ 𝑧 ,

̂

̂

𝐹 𝐽̂𝐴 1−𝑒 ̂ 𝑅𝐹 +2𝛾

One should note that, since be obtained ( −(𝑅̂ +2𝛾)𝑡 𝑒

𝜔0 𝜎̂ + 𝑔(𝑎̂𝜎̂ + + 𝑎̂† 𝜎̂ − ), (4) 2 𝑧 where 𝜔 is the field frequency, 𝜔0 the atomic transition frequency, 𝑔 is the atom–field interaction constant, 𝑎̂† and 𝑎̂ are the creation and annihilation operators, respectively, 𝜎̂ + and 𝜎̂ − are the raising and lowering atomic operators. Furthermore, the operator 𝑛̂ = 𝑎̂† 𝑎̂ is the so-called number operator and 𝜎̂ 𝑧 = [𝜎̂ + , 𝜎̂ − ] is the Pauli-spin matrix related to the atomic inversion. In the above equation the cavity and atomic decay terms are given by

𝐹 𝐽̂𝐴 1−𝑒 ̂ 𝑅𝐹 +2𝛾

𝜌(𝑡) ̂ =

1 + 𝐽̂𝐴

̂

× exp

𝑡

∫0

𝑒

̂ 𝐿̂ 𝐴 +𝐿̂ 𝐹 )𝑡′ −(𝑅+

(7)

(𝐽̂𝐴 + 𝐽̂𝐹 )𝑒

)

𝑑𝑡′ 𝜌(0), ̂

−(𝑅𝐴 +2𝜅)𝑡 𝐽̂𝐹 1−𝑒 ̂ 𝑅𝐴 +2𝜅

̂

̂

𝜌(0), ̂

̂ 1 − 𝑒−(𝑅𝐹 +2𝛾)𝑡

𝑅̂ 𝐹 + 2𝛾

̂

𝑒

−(𝑅𝐴 +2𝜅)𝑡 𝐽̂𝐹 1−𝑒 ̂ 𝑅𝐴 +2𝜅

𝜌(0). ̂

(19)

∞ ( ) ∑ = 𝑇 𝑟 𝜌̂1 (𝑡)𝜎̂ 𝑥 = ⟨𝑒, 𝑛|𝜌̂1 (𝑡)𝜎̂ 𝑥 |𝑒, 𝑛⟩ + ⟨𝑔, 𝑛|𝜌̂1 (𝑡)𝜎̂ 𝑥 |𝑔, 𝑛⟩ 𝑛=0

=

∞ ∑

∞ ∑ (2𝜅𝜁 ⋆ )𝑚 1 sin(2𝜃)𝑒−𝛾𝑡 𝑒−2𝜂𝑛𝑡 4 𝑚! 𝑛=0 𝑚=0

̂ 𝑎̂†𝑚 |𝑛⟩ + c.c. × ⟨𝑛|𝑎̂𝑚 𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼) (20) ( ) where 𝜂 = 𝜅 + 𝑖𝜒 and 𝑇 𝑟 𝜌̂2 (𝑡)𝜎̂ 𝑥 = 0. In the above relation, 𝜌̂1 (𝑡) and 𝜌̂2 (𝑡) have been defined in (A.11) and (B.6), respectively. Eq. (20) can be simplified as ⟨𝜎̂ 𝑥 ⟩ =

(9)

̂ 𝐿̂ 𝐴 +𝐿̂ 𝐹 )𝑡′ (𝑅+

̂

At this stage, to measure the quasiprobability distributions, we use the method that has been proposed in Ref. [29]. To achieve this purpose, one should measure the atomic polarization ⟨𝜎̂ 𝑥 ⟩ as follows ( ) ⟨𝜎̂ 𝑥 ⟩ = 𝑇 𝑟 𝜌(𝑡) ̂ 𝜎̂ 𝑥

̂ 𝐿̂ 𝐴 +𝐿̂ 𝐹 )𝑡 (𝑅+

(

̂

3. Measuring quasiprobability distribution functions

with 𝜒 = as the new interaction constant. Since 𝜒 ≪ 𝑔, decay effects become more important in this regime. With this effective Hamiltonian, we write the following master equation for the so-called dispersive interaction

𝜌(𝑡) ̂ = 𝑒

(17)

𝜌(𝑡), ̂

In Appendices A and B we give the explicit expressions for the above density operators.

− 𝜔2 )

𝑑 𝜌̂ = −𝑖𝜒[𝑛̂ 𝜎̂ 𝑧 , 𝜌] ̂ + ̂ 𝐹 𝜌̂ + ̂ 𝐴 𝜌. ̂ 𝑑𝑡 We may solve (9) by doing

)

(18)

𝜌̂2 (𝑡) = 𝑒𝑡(𝑅+𝐿𝐴 +𝐿𝐹 ) 𝐽̂𝐴

(8) 2 0 ∕(𝜔0

̂

̂

̂

where 𝛾 is the atomic decay rate and 𝜅 is the cavity-field decay rate. We consider the strong interaction regime, where 𝑔 ≫ 𝛾, 𝜅. Moreover, the dispersive regime where |𝜔 − 𝜔0 | ≫ 𝑔 is studied such that one can obtain an effective Hamiltonian (for instance, via a small rotation [35]) that reads [7]

2𝑔 2 𝜔

(16)

𝜌(0). ̂

= 0, the following simplification may

1 − 𝑒−(𝑅𝐹 +2𝛾)𝑡 𝑅̂ 𝐹 + 2𝛾

𝜌̂1 (𝑡) = 𝑒𝑡(𝑅+𝐿𝐴 +𝐿𝐹 ) 𝑒

and

𝐻̂ 𝑒𝑓 𝑓 = 𝜒 𝑛̂ 𝜎̂ 𝑧

𝐽̂𝐴2 𝜌(𝑡) ̂

where

(6)

𝐿̂ 𝐹 𝜌̂ = −𝜅(𝑎̂† 𝑎̂𝜌̂ + 𝜌̂𝑎̂† 𝑎), ̂

−(𝑅𝐴 +2𝜅)𝑡 𝐽̂𝐹 1−𝑒 ̂ 𝑅𝐴 +2𝜅

𝜌(𝑡) ̂ = 𝜌̂1 (𝑡) + 𝜌̂2 (𝑡),

in which the subscripts ‘‘A’’ and ‘‘F’’ refer to the terms atom and field, respectively. The corresponding superoperators are defined as

𝐿̂ 𝐴 𝜌̂ = −𝛾(𝜎̂ + 𝜎̂ − 𝜌̂ + 𝜌̂𝜎̂ + 𝜎̂ − ),

𝑒

and therefore,

(5)

𝐽̂𝐹 𝜌̂ = 2𝜅 𝑎̂𝜌̂𝑎̂ ,

̂

−(𝑅̂ +2𝛾)𝑡

̂

𝜌(𝑡) ̂ = 𝑒(𝑅+𝐿𝐴 +𝐿𝐹 )𝑡 𝑒

𝐻̂ = 𝜔𝑛̂ +

𝐽̂𝐴 𝜌̂ = 2𝛾 𝜎̂ − 𝜌̂𝜎̂ + ,

(15)

which commute with all the other superoperators involved, such that the solution of the evolved density matrix is written as

𝑑 𝜌̂ ̂ 𝜌] = −𝑖[𝐻, ̂ + ̂ 𝐹 𝜌̂ + ̂ 𝐴 𝜌, ̂ (3) 𝑑𝑡 with 𝐻̂ as the atom–field interaction Hamiltonian given by (we set ℏ = 1)

†

(14)

have been used. In the above equations, we have defined the superoperators 𝑅̂ 𝐹 and 𝑅̂ 𝐴 as

We consider the interaction of a two-level atom and a quantized single-mode field in a high-𝑄, non-ideal, cavity, i.e., the single-mode field may lose photons through the cavity mirrors. Besides this dissipative process, we also consider that the (excited) atom can spontaneously emit photons into non-cavity modes. In the Markovian approximation we can include dissipation into the master equation for the reduced density matrix operator, 𝜌, ̂ as [30–34]

𝑗 = 𝐴, 𝐹

(13)

and

2. Dispersive interaction between a two-level atom and a quantized field

̂ 𝑗 𝜌̂ = (𝐽̂𝑗 + 𝐿̂ 𝑗 )𝜌, ̂

(12)

∞ ∞ ∑ ∑ (2𝜅𝜁 ⋆ )𝑚 (𝑛 + 𝑚)! 1 sin(2𝜃)𝑒−𝛾𝑡 𝑒−2𝜂𝑛𝑡 4 𝑚! 𝑛! 𝑛=0 𝑚=0

̂ × ⟨𝑛 + 𝑚|𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)|𝑛 + 𝑚⟩ + c.c.

(21)

𝜅 (1 − 𝑒−2𝜂𝑡 ), 𝜂

(22)

Since 2𝜅𝜁 ⋆ =

(10) 70

N. Yazdanpanah et al.

Optics Communications 400 (2017) 69–73

zero, tan 𝜙 is zero. We take in particular the interaction time of the first zero, which gives 𝜙 = 𝜋 and define 𝑠 = 𝜇+1 to write 𝜇−1 ⟨𝜎̂ 𝑥 ⟩ =

∞ ( ) ∑ 1 𝑠+1 𝑛 ̂ ⟨𝑘|𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)|𝑘⟩, sin(2𝜃)𝑒−𝛾𝑡 2 𝑠 − 1 𝑘=0

(29)

or finally, comparing to Eq. (2) ⟨𝜎̂ 𝑥 ⟩ =

(1 − 𝑠)𝜋 sin(2𝜃)𝑒−𝛾𝑡 𝐹 (𝛼, 𝑠), 2

(30)

i.e., by measuring an atomic observable, namely, the atomic polarization, we can reconstruct a quasiprobability distribution function even though atomic and field decays take place. 4. Conclusions We have solved the dispersive interaction between a quantized electromagnetic field and a decaying two-level atom in cavity subject to losses by using superoperator techniques. We have shown that even in the both decaying cases we can still obtain information about the initial cavity field by means of 𝑠-parametrized quasiprobability distribution functions. Due to the fact that these functions contain complete information about the state of the cavity field, we are able to determine the field state, completely. One thing to consider is the fact that an effective (dispersive) interaction produces much slower processes such that, both atomic and field decays, may be of importance. Moreover, if we consider a very small 𝜃, i.e., the atom being mostly in the ground state with a very small contribution from the excited state, the reconstruction is still possible, as

Fig. 1. Plot of the function 𝜖 + 𝑒−2𝜅𝑡 (sin 𝜒𝑡 − 𝜖 cos 𝜒𝑡) as a function of 𝜒𝑡 for 𝜅 = 0.05 and 𝜖 = 0.05 (solid line) and 𝜅 = 0.1 and 𝜖 = 0.1 (dashed line).

we can obtain ⟨𝜎̂ 𝑥 ⟩ =

∞ [ ∞ ]𝑚 ∑ ∑ 𝜅 1 𝑒−2𝜂𝑛𝑡 sin(2𝜃)𝑒−𝛾𝑡 (1 − 𝑒−2𝜂𝑡 ) 4 𝜂 𝑚=0 𝑛=0

(𝑛 + 𝑚)! ⟨𝑛 + 𝑚|𝜌̂𝐹 (0)|𝑛 + 𝑚⟩ + c.c. 𝑚!𝑛!

×

(23)

By changing the summation index in the second sum of the above equation with 𝑛 + 𝑚 = 𝑘, one may obtain

⟨𝜎̂ 𝑥 ⟩ ≈ (1 − 𝑠)𝜋𝜃𝑒−𝛾𝑡 𝐹 (𝛼, 𝑠),

∞[ ∞ ]𝑘−𝑛 ∑ ∑ 𝜅 1 𝑒−2𝜂𝑛𝑡 (1 − 𝑒−2𝜂𝑡 ) ⟨𝜎̂ 𝑥 ⟩ = sin(2𝜃)𝑒−𝛾𝑡 4 𝜂 𝑘=𝑛 𝑛=0

𝑘! ̂ ⟨𝑘|𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)|𝑘⟩ + c.c. (𝑘 − 𝑛)!𝑛!

×

and, although the reconstruction would be severely diminished by such a small value of 𝜃 and the fact that we are considering a finite atomic decay rate, it is still possible to obtain whole information from the initial field state. Finally note that, if we consider 𝜅 = 0, i.e., an ideal cavity, 𝜇 = 1 [see Eq. (28)] such that the Wigner distribution function may be reconstructed.

(24)

It is noticeable that Eq. (24) can be reduced to equation (20) of Ref. [29] via choosing 𝛾 = 0 (without considering the atomic decay) and 𝜃 = 𝜋4 (especial case) as is expected. We now start the second sum from 𝑘 = 0 (as we would add only zeros) and change the order of the sums to obtain ⟨𝜎̂ 𝑥 ⟩ =

Appendix A. Calculation of 𝝆̂𝟏 (𝒕)

∞ [ ]𝑘 ∑ 1 𝜅 ̂ sin(2𝜃)𝑒−𝛾𝑡 (1 − 𝑒−2𝜂𝑡 ) ⟨𝑘|𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)|𝑘⟩ 4 𝜂 𝑘=0

×

∞ ∑ 𝑛=0

𝑒−2𝜂𝑛𝑡

𝑘! + c.c. ( )𝑛 (𝑘 − 𝑛)!𝑛! 𝜅 −2𝜂𝑡 ) (1 − 𝑒 𝜂

In order to calculate the evolved density matrix (18) it is needed to find both terms on the right hand side of that equation. For this, we consider the atom and field to be initially in arbitrary states such that the initial atom–field density operator reads as

(25)

The second sum can be cut at 𝑛 = 𝑘 as the terms for 𝑛 > 𝑘 are zero. Therefore, it may be summed to give ⟨𝜎̂ 𝑥 ⟩ =

∞ ( ∑

1 sin(2𝜃)𝑒−𝛾𝑡 4 𝑘=0

𝜅 + 𝑖𝜒𝑒−2𝜂𝑡 )𝑘 𝜂

̂ ⟨𝑘|𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)|𝑘⟩ + c.c.

̂ 𝜌(0) ̂ = 𝜌̂𝐴 (0)𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼),

𝜇=

|𝜓𝐴 (0)⟩ = sin 𝜃|𝑒⟩ + cos 𝜃|𝑔⟩,

𝜖 + 𝑒−2𝜅𝑡 (sin 𝜒𝑡 − 𝜖 cos 𝜒𝑡) , 𝜖 2 + 𝑒−2𝜅𝑡 (cos 𝜒𝑡 + 𝜖 sin 𝜒𝑡)

(27)

with 𝜖 = 𝜅∕𝜒 we finally obtain ∞ ∑ 1 ̂ ⟨𝜎̂ 𝑥 ⟩ = sin(2𝜃)𝑒−𝛾𝑡 𝜇𝑛 cos 𝑛𝜙⟨𝑘|𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)|𝑘⟩. 2 𝑘=0

(A.2)

where 𝜃 is an arbitrary angle and |𝑒⟩ (|𝑔⟩) is the excited (ground) state of the atom. We have considered an initially displaced (by an amplitude 𝛼) arbitrary field. We rewrite Eq. (A.1) with the help of the atomic operators as ( ) ̂ 𝜌(0) ̂ = sin2 𝜃 𝜎̂ 𝑧 + 𝜎̂ − 𝜎̂ + + sin(2𝜃) 𝜎̂ 𝑥 𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼), (A.3)

𝜖 2 + 𝑒−4𝜅𝑡 + 2𝜖 sin 𝜒𝑡𝑒−2𝜅𝑡 , 1 + 𝜖2

tan 𝜙 = −

(A.1)

in which 𝜌̂𝐴 (0) = |𝜓𝐴 (0)⟩⟨𝜓𝐴 (0)| with

(26)

By defining √

(31)

where 𝜎̂ 𝑥 = (𝜎̂ + + 𝜎̂ − )∕2. By acting the superoperator 𝑅̂ 𝐴 on the atomic operators one arrives at

(28)

Now, we plot in Fig. 1 the numerator of tan 𝜙, i.e., the function 𝜖 + 𝑒−2𝜅𝑡 (sin 𝜒𝑡−𝜖 cos 𝜒𝑡) for two values of 𝜅. Whenever this function crosses 71

𝑅̂ 𝐴 𝜎̂ 𝑧 = 0,

𝑅̂ 𝐴 𝜎̂ − 𝜎̂ + = 0,

𝑅̂ 𝐴 𝜎̂ 𝑥 = −2𝜒 𝜎̂ 𝑦 ,

𝑅̂ 𝐴 𝜎̂ 𝑦 = 2𝜒 𝜎̂ 𝑥 .

(A.4)

N. Yazdanpanah et al.

Optics Communications 400 (2017) 69–73

Appendix B. Calculation of 𝝆̂𝟐 (𝒕)

In order to find the evolved density matrix, we need to calculate 𝜌̂1 (𝑡) in (19), so we have ̂

𝑒

−(𝑅𝐴 +2𝜅)𝑡 𝐽̂𝐹 1−𝑒 ̂ 𝑅𝐴 +2𝜅

𝜌(0) ̂ =

Now, we pay our attention to the calculation of 𝜌̂2 (𝑡). At first, we note the following facts

∞ ) ( ̂ ∑ 1 − 𝑒−(𝑅𝐴 +2𝜅)𝑡 𝑚 1 ̂𝑚 ̂ † ̂ 𝜌̂𝐴 (0) 𝐽𝐹 𝐷 (𝛼)𝜌̂𝐹 (0)𝐷(𝛼) 𝑚! 𝑅̂ 𝐴 + 2𝜅 𝑚=0

∞ ∑ 1 ̂ 𝑎̂†𝑚 = (2𝜅)𝑚 𝑎̂𝑚 𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼) 𝑚! 𝑚=0 ( 𝑡 ) ′ 𝑚 ̂ 𝑑𝑡′ 𝑒−(𝑅𝐴 +2𝜅)𝑡 × 𝜌̂𝐴 (0), ∫0

𝐽̂𝐴 𝜎̂ ± = 𝐽̂𝐴 𝜎̂ − 𝜎̂ + = 0, 𝐽̂𝐴 𝜎̂ 𝑧 = 2𝛾 𝜎̂ − 𝜎̂ + = 2𝛾|𝑔⟩⟨𝑔|. Therefore, we have

(A.5)

̂

𝐽̂𝐴

with 𝑡

∫0

𝑡

′

̂

𝑑𝑡′ 𝑒−(𝑅𝐴 +2𝜅)𝑡 𝜌̂𝐴 (0) = 𝑡

=

′

∫0

′

𝑡

𝑑𝑡′ 𝑒

=

−2𝜅𝑡′

′

′

𝑑𝑡′ 𝑒

∞ 𝑙 ̂ ∑ 1 − 𝑒−(𝑅𝐹 +2𝛾)𝑡 ∑ 𝑡𝑙+1 𝑙! = (−1)𝑙 (2𝛾)𝑙−𝑘 𝑅̂ 𝑘𝐹 , ̂ (𝑙 + 1)! 𝑘!(𝑙 − 𝑘)! 𝑅𝐹 + 2𝛾 𝑙=0 𝑘=0

cos(2𝜒𝑡′ )

𝑒

𝑖𝜒 𝜎̂ 𝑧 𝑡′

𝜎̂ 𝑥 𝑒

𝑡′

𝑒−𝑖𝜒 𝜎̂ 𝑧 𝜎̂ 𝑦 𝑒𝑖𝜒 𝜎̂ 𝑧

𝑡′

𝜌̂2 (𝑡) = 2𝛾sin2 𝜃 (A.7)

𝜌(0) ̂ =

∞ ∑ 1 ̂ 𝑎̂†𝑚 (2𝜅)𝑚 𝑎̂𝑚 𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼) 𝑚! 𝑚=0 { ( ) ( ) 1 − 𝑒−2𝜅𝑡 𝑚 × sin2 𝜃 𝜎̂ 𝑧 + 𝜎̂ − 𝜎̂ + 2𝜅 } ( ) 1 + sin(2𝜃) 𝜎̂ + 𝜁 ⋆𝑚 + 𝜎̂ − 𝜁 𝑚 , 2

(A.8)

×

{ (

1 − 𝑒−2𝜅𝑡 2𝜅

[1] K. Vogel, H. Risken, Phys. Rev. A 40 (1989) 2847. [2] U. Leonhardt, Measuring the Quantum State of Light, Cambridge University Press, Cambridge, 1997. [3] A. Zucchetti, W. Vogel, M. Tasche, D.G. Welsch, Phys. Rev. A 54 (1996) 1678. [4] S. Wallentowitz, W. Vogel, Phys. Rev. A 53 (1996) 4528. [5] Z. Kis, T. Kiss, J. Janszky, P. Adam, S. Wallentowitz, W. Vogel, Phys. Rev. A 59 (1999) R39. [6] K. Banaszek, K. Wodkiewcz, Phys. Rev. Lett. 76 (1996) 4344. [7] L.G. Lutterbach, L. Davidovich, Phys. Rev. Lett. 78 (1997) 2547. [8] H.M. Moya-Cessa, S.M. Dutra, J.A. Roversi, A. Vidiella-Barranco, J. Modern Opt. 46 (1999) 555. [9] H.M. Moya-Cessa, F. Soto-Eguibar, J.M. Vargas-Martínez, R. Juárez-Amaro, A. Zuniga-Segundo, Phys. Rep. 513 (2012) 229. [10] B. Swingle, I.H. Kim, Phys. Rev. Lett. 113 (2014) 260501. [11] R.P. Rundle, T. Tilma, J.H. Samson, M.J. Everitt, 1605, arXiv:160508922v3 [quantph]. [12] D. Lv, S. An, M. Um, J. Zhang, J.-N. Zhang, M.S. Kim, K. Kim, Phys. Rev. A 95 (2017) 043813. [13] M.V. Satyanarayana, P. Rice, R. Vyas, H.J. Carmifchael, J. Opt. Soc. Amer. B Opt. Phys. 6 (1989) 228. [14] H.M. Moya-Cessa, A. Vidiella-Barranco, J. Modern Opt. 39 (1992) 2481. [15] H.M. Moya-Cessa, J.A. Roversi, S.M. Dutra, A. Vidiella-Barranco, Phys. Rev. A 60 (1999) 4029. [16] U. Leonhardt, H. Paul, Phys. Rev. A 48 (1993) 4598. [17] U. Leonhardt, H. Paul, J. Modern Opt. 41 (1994) 1427. [18] G.M. D’Ariano, M. de Laurentis, M.G.A. Paris, A. Porzio, S. Solimeno, J. Opt. B 4 (2002) S127. [19] A. Royer, Phys. Rev. A 53 (1996) 70. [20] A. Wünsche, Quant. Opt.: J. Euro. Opt. Soc. 3 (1991) 359. [21] H.M. Moya-Cessa, P.L. Knight, Phys. Rev. A 48 (1993) 2479. [22] K. Husimi, Phys. Math. Soc. Jpn. 22 (1940) 264. [23] E.P. Wigner, Phys. Rev. 40 (1932) 749.

(A.10)

∞ ∑ 1 ̂ 𝑎̂†𝑚 (2𝜅)𝑚 𝑎̂𝑚 𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼) 𝑚! 𝑚=0

)𝑚 ( ) sin2 𝜃 𝜎̂ 𝑧 + 𝜎̂ − 𝜎̂ +

} ( ) 1 ⋆𝑚 𝑚 ̂ 𝜎̂ 𝑧 𝑛)𝑡 ̂ + sin(2𝜃) 𝜎̂ + 𝜁 + 𝜎̂ − 𝜁 𝑒−(𝛾 𝜎̂ + 𝜎̂ − +𝜅 𝑛−𝑖𝜒 . 2

(B.6)

References

we finally arrive at the end of our first task, namely the determination of 𝜌̂1 (𝑡) ̂ 𝜎̂ 𝑧 𝑛)𝑡 ̂ 𝜌̂1 (𝑡) = 𝑒−(𝛾 𝜎̂ + 𝜎̂ − +𝜅 𝑛+𝑖𝜒

𝑙 ∑ 𝑡𝑙+1 𝑙! (−1)𝑙 (2𝛾)𝑙−𝑘 (𝑙 + 1)! 𝑘!(𝑙 − 𝑘)! 𝑘=0

× 𝑒(−𝑖𝜒−𝜅)𝑛𝑡̂ |𝑔⟩⟨𝑔|.

= 𝜎̂ 𝑥 cos(2𝜒𝑡 ) + 𝜎̂ 𝑦 sin(2𝜒𝑡 ),

̂

(B.5)

̂ 𝑎̂†𝑚 𝑛̂ 𝑗 × 𝑒(𝑖𝜒−𝜅)𝑛𝑡̂ 𝑛̂ 𝑘−𝑗 𝑎̂𝑚 𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)

Utilizing (A.5) and (A.9) results in the expression −(𝑅𝐴 +2𝜅)𝑡 𝐽̂𝐹 1−𝑒 ̂ 𝑅𝐴 +2𝜅

∞ ∑ 𝑙=0

By using 𝜎̂ 𝑥 = (𝜎̂ + + 𝜎̂ − )∕2 and 𝜎̂ 𝑦 = (𝜎̂ + − 𝜎̂ − )∕2𝑖 and defining 𝜁 = (𝜒, 𝜅, 𝑡) + 𝑖(𝜒, 𝜅, 𝑡) one can deduce ( 𝑡 ) ( ) ′ 1 − 𝑒−2𝜅𝑡 𝑚 ̂ 𝑑𝑡′ 𝑒−(𝑅𝐴 +2𝜅)𝑡 𝑚 𝜌̂𝐴 (0) = ∫0 2𝜅 ( ) { } 1 × sin2 𝜃 𝜎̂ 𝑧 + 𝜎̂ − 𝜎̂ + + sin(2𝜃) 𝜎̂ + 𝜁 ⋆𝑚 + 𝜎̂ − 𝜁 𝑚 . (A.9) 2

𝑒

𝑘! 𝑛̂ 𝑘−𝑗 𝜌̂𝐹 𝑛̂ 𝑗 , 𝑗!(𝑘 − 𝑗)!

∞ 𝑘 ∑ ∑ (1 − 𝑒−2𝜅𝑡 )𝑚 𝑘! (2𝑖𝜒)𝑘 (−1)𝑗 × 𝑚! 𝑗!(𝑘 − 𝑗)! 𝑚=0 𝑗=0

′

= 𝜎̂ 𝑦 cos(2𝜒𝑡′ ) − 𝜎̂ 𝑥 sin(2𝜒𝑡′ ).

(B.4)

and finally

−2𝜅𝑡′

′

𝑘 ∑ (−1)𝑗 𝑗=0

In obtaining Eq. (A.6) we have used the following relations −𝑖𝜒 𝜎̂ 𝑧 𝑡′

(B.3)

where we have used the binomial theorem. For the superoperator 𝑅̂ 𝑘𝐹 we have

(A.6)

𝑅̂ 𝑘𝐹 𝜌̂𝐹 = (2𝑖𝜒)𝑘

𝑑𝑡′ 𝑒 sin(2𝜒𝑡′ ) ∫0 −2𝜅 sin(2𝜒𝑡) − 2𝜒 cos(2𝜒𝑡) −2𝜅𝑡 2𝜒 = 𝑒 + . 4𝜒 2 + 4𝜅 2 4𝜒 2 + 4𝜅 2

(𝜒, 𝜅, 𝑡) =

(B.2)

so that,

∫0 −2𝜅 cos(2𝜒𝑡) + 2𝜒 sin(2𝜒𝑡) −2𝜅𝑡 2𝜅 𝑒 + , = 4𝜒 2 + 4𝜅 2 4𝜒 2 + 4𝜅 2 𝑡

∞ ∑ (1 − 𝑒−2𝜅𝑡 )𝑚 𝑚! 𝑚=0

∞

where we have defined the following abbreviations (𝜒, 𝜅, 𝑡) =

𝜌(0) ̂ = 2𝛾sin2 𝜃

̂ ∑ 1 − 𝑒−𝑥𝑡 𝑡𝑙+1 = , (−𝑥) ̂ 𝑙 𝑥̂ (𝑙 + 1)! 𝑙=0

{ sin2 𝜃 𝜎̂ 𝑧 + 𝜎̂ − 𝜎̂ +

−2𝜅𝑡′

−(𝑅𝐴 +2𝜅)𝑡 𝐽̂𝐹 1−𝑒 ̂ 𝑅𝐴 +2𝜅

Also, for each operator 𝑥̂ one can obtain

′

∫0 ( )} + sin(2𝜃) 𝜎̂ 𝑥 cos(2𝜒𝑡′ ) + 𝜎̂ 𝑦 sin(2𝜒𝑡′ ) ( ) 1 − 𝑒−2𝜅𝑡 sin2 𝜃 𝜎̂ 𝑧 + 𝜎̂ − 𝜎̂ + = 2𝜅 { } + sin(2𝜃) 𝜎̂ 𝑥 (𝜒, 𝜅, 𝑡) + 𝜎̂ 𝑦 (𝜒, 𝜅, 𝑡) ,

𝑡

𝑒

̂ 𝑎̂†𝑚 |𝑔⟩⟨𝑔|. × 𝑎̂𝑚 𝐷̂ † (𝛼)𝜌̂𝐹 (0)𝐷(𝛼)

𝑑𝑡′ 𝑒−2𝜅𝑡 𝑒−𝑅𝐴 𝑡 𝜌̂𝐴 (0)

𝑑𝑡′ 𝑒−2𝜅𝑡 𝑒−𝑖𝜒 𝜎̂ 𝑧 𝑡 𝜌̂𝐴 (0) 𝑒𝑖𝜒 𝜎̂ 𝑧 𝑡

∫0

̂

(B.1)

(A.11)

72

N. Yazdanpanah et al.

Optics Communications 400 (2017) 69–73 [29] [30] [31] [32] [33] [34] [35]

[24] [25] [26] [27]

R.J. Glauber, Phys. Rev. A 131 (1963) 2766. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277. F.A.M. de Oliveira, M.S. Kim, P.L. Knight, V. Buzek, Phys. Rev. A 41 (1990) 2645. D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 77 (1996) 4281. [28] P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 89 (2002) 200402.

73

R. Juarez-Amaro, H.M. Moya-Cessa, Phys. Rev. A 68 (2003) 023802. T. Quang, P.L. Knight, V. Buzek, Phys. Rev. A 44 (1991) 6092. H.J. Carmichael, Phys. Rev. Lett. 55 (1985) 2790. C.M. Savage, Phys. Rev. Lett. 60 (1988) 1828. C.M. Savage, H.J. Carmichael, IEEE J. Quantum Electron. 24 (1988) 1495. S.-Y. Zhu, L.Z. Wang, H. Fearn, Phys. Rev. A 44 (1991) 737. A.B. Klimov, L.L. Sanchez-Soto, Phys. Rev. A 61 (2000) 063802.