coherent operation of photon subtraction and addition for squeezed thermal states: analysis of...

Coherent operation of photon subtraction andaddition for squeezed thermal states: analysis

of nonclassicality and decoherence

Shuai Wang,1,* Xue-xiang Xu,2 Hong-chun Yuan,3 Li-yun Hu,2 and Hong-yi Fan1

1Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China2College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China

3College of Optoelectronic Engineering, Changzhou Institute of Technology, Changzhou 213002, China*Corresponding author: [email protected]

Received April 13, 2011; revised June 5, 2011; accepted July 2, 2011;posted July 12, 2011 (Doc. ID 145875); published August 15, 2011

We study nonclassical properties of the optical field when photon subtraction/addition coherent operationtaþ ra† (jtj2 þ jrj2 ¼ 1) acts on a squeezed thermal state (STS) by examining its quadrature squeezing, sub-Poissonian statistics, and negativity of the Wigner function. The degree of squeezing of the coherent operatedsqueezed thermal state (COSTS) becomes weaker. The Mandel Q parameter decreases with r and quickly becomesnegative, which implies sub-Poissonian statistics. Both the negative dip and negative area of the Wigner functionincrease with r for small squeezing values of κ. Decoherence of the COSTS in an amplitude-damping channel isstudied by the time evolution of the Wigner function. The length of time that this nonclassical field preserves itspartial negativity of the Wigner function can be modulated by the coherent operation. All these results indicatethat the nonclassicality of the STS is sensitive to the coherent operation. © 2011 Optical Society of America

OCIS codes: 270.0270, 270.5290.

1. INTRODUCTIONSqueezing is a main resource for various important protocolsin quantum information [1–3]. In particular, non-Gaussiansqueezed states with strongly nonclassical properties, such asnegativity of quasi-probability phase space distributions andentanglement, may constitute powerful resources for the effi-cient implementation of quantum communication and compu-tation [1–6]. Therefore, some theoretical and experimentalefforts have been made toward engineering and controllinghighly nonclassical, non-Gaussian squeezed states of opticalfields [7–16]. Performing photon subtraction on a given Gaus-sian state is a possible approach to generate non-Gaussianstates [17,18]. For example, the photon-subtracted squeezedstates [7], and photon-subtracted-then-added thermal stateshave both been generated in experiments [12]. Another exam-ple is the photon-subtracted squeezed vacuum state whosenonclassicality was first studied by Biswas and Agarwal[19], and further studied by Hu and Fan [14] by examiningits Wigner function. In Ref. [15], Hu et al. came up with thephoton-subtracted squeezed thermal state (STS) and investi-gated its nonclassicality by discussing the negativity of theWigner function. The photon addition is known to create anonclassical state from any classical one [20], and both thephoton-subtracted [10] and the photon-added squeezed states[3,4,13] were suggested to improve the fidelity of continuousvariable quantum information processing.

Since single-photon subtraction and addition have beensuccessfully demonstrated experimentally [21–23], it is inter-esting to explore whether these elementary operations canbe combined to engineer complicated quantum states of op-tical fields. Kim et al. [24] proposed a combined photon sub-traction and addition experiment scheme to probe quantum

commutation rules ½a; a†� ¼ 1. Fiurášek [25] later proposeda scheme for the approximate probabilistic realization of anarbitrary operation that can be expressed as a function ofphoton number operator a†a. Recently, Lee and Nha consid-ered a coherent superposition of photon addition and subtrac-tion, taþ ra†, (jtj2 þ jrj2 ¼ 1) acting on a coherent state anda thermal state to form non-Gaussian states [26], and investi-gated the emerging nonclassical properties via the Wignerfunction. They also proposed an experimental scheme toimplement this coherent operation. The production of non-Gaussian states has inspired broad interest because of theirstrongly nonclassical properties [10,27–29]. In this paper, wedemonstrate how the nonclassicality is enhanced and how thesqueezing varies when the coherent operation acts on an STS.Since a dissipative quantum channel tends to deteriorate thedegree of nonclassicality, we further explore the decoherenceof the coherent operated squeezed thermal state (COSTS) inthe amplitude-damping channel through the dynamic evolu-tion of partial negativity of the Wigner function.

The paper is organized as follows. In Section 2, based on thenormal ordering form of the density operator of a STS, we de-rive the normalization constant of the COSTS. In Section 3,two observable nonclassical effects, quadrature squeezingand Mandel Q parameter for this state, are calculated analy-tically. In Section 4, in terms of negative properties of Wignerfunctions, we study in detail how the coherent operation ofphoton subtraction and addition enhances the nonclassicality.Then, in Section 5, we derive an analytical expression of thetime evolution of the Wigner function for the COSTS in theamplitude-damping channel and discuss its loss of nonclassi-cality. The main results are summarized in Section 6.

Wang et al. Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B 2149

0740-3224/11/092149-10$15.00/0 © 2011 Optical Society of America

2. NORMALIZATION OF THE COSTSAfter an STS is operated by the coherent operation taþ ra†

with jtj2 þ jrj2 ¼ 1, where a† and a are creation and annihila-tion operators, the density operator of the resulting stateCOSTS is

ρ ¼ N−1ðtaþ ra†Þρsðt�a† þ r�aÞ; ð1Þ

where N ¼ Trfðtaþ ra†Þρsðt�a† þ r�aÞg is introduced fornormalizing ρ, and ρs is the normalized density operator ofthe STS

ρs ¼ SðκÞρthS†ðκÞ; ð2Þ

where SðκÞ ¼ exp½κða†2 − a2Þ=2� is the squeezing operator[30,31] with squeezing parameter κ, and ρth ¼ ð1 − e−βÞe−βa†ais the density operator of the thermal (chaotic) state, withthe average photon number �n ¼ 1

eβ−1[32]. For simplifying the

calculation, we employ the normal ordering form of ρs [33,34]

ρs ¼1

τ1τ2: exp

�−X2

2τ21−P2

2τ22

�:; ð3Þ

where : : denotes the normal ordering, X ¼ ða† þ aÞ= ffiffiffi2

pand P ¼ ða − a†Þ=ð ffiffiffi

2p

iÞ, 2τ21 ¼ ð2�nþ 1Þe2κ þ 1, and 2τ22 ¼ð2�nþ 1Þe−2κ þ 1, which lead to the following relations:

τ21 − τ22 ¼ ð2�nþ 1Þ sinh 2κ; ð4Þ

τ21 þ τ22 ¼ 2ð�n cosh 2κ þ cosh2κÞ; ð5Þ

τ21τ22 ¼ �n2 þ ð2�nþ 1Þ cosh2 κ: ð6Þ

Using the completeness relation of coherent stateRd2zπ jzihzj ¼ 1 [35] and the following integral formula [36],

Zd2zπ expðζjzj2 þ ξzþ ηz� þ f z2 þ gz�2Þ

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiζ2 − 4f g

p exp

�−ζξηþ ξ2gþ η2f

ζ2 − 4f g

�; ð7Þ

with convergent condition Reðξ� f � gÞ < 0 and Reðζ2−4f gξ�f�gÞ <

0, we obtain the normalization constant for ρ [see Appendix A]:

N ¼ τ21 þ τ22 − 22

jtj2 þ ðτ21 − τ22Þ2

ðtr� þ rt�Þ þ τ21 þ τ222

jrj2: ð8Þ

Substituting Eqs. (4)–(6) into Eq. (8), we see

N ¼ jrj2 þ ð2�nþ 1Þ sinh 2κ2

ðtr� þ rt�Þ þ ð2�nþ 1Þ cosh 2κ − 12

:

ð9ÞWhen κ ¼ 0, Eq. (9) reduces to N ¼ �nþ jrj2, which is

just the normalization constant for the coherent operatedthermal state [26]; when t ¼ 1, Eq. (9) reduces to N ¼�n cosh 2κ þ sinh2 κ, the normalization constant of a single-photon-subtracted STS [15]; while r ¼ 1, Eq. (9) reduces toN ¼ �n cosh 2κ þ cosh2 κ, the case of a single-photon-addedSTS [16].

3. OBSERVABLE NONCLASSICAL EFFECTSIn this section, we investigate two observable nonclassicaleffects: quadrature squeezing and sub-Poissonian statisticsof the COSTS.

We first explore how the squeezing quantitatively changeswhen the coherent operation is acted on an STS. For this pur-pose, we consider the quadrature operator Xθ ¼ ae−iθ þ a†eiθ.The squeezing is characterized by the minimum valuehΔ2Xθimin < 1 with respect to θ or by the normal orderingform h: Δ2Xθ :imin < 0 [37]. Upon expanding the terms ofh: Δ2Xθ :i, one can minimize its value over the whole angleθ, which is then given by [26,37]

Sopt ≡ h: Δ2Xθ :imin ¼ −2jha†2i − ha†i2j þ 2ha†ai − 2jha†ij2;ð10Þ

whose negative value in the range ½−1; 0Þ implies squeezing(or nonclassical). For the STS ρs, the degree of squeezing is

S0 ¼ ð2�nþ 1Þe−2κ − 1; ð11Þ

so the negative value of S0 emerges only when κ >ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2�nþ 1Þp[38–40], and the maximal degree of squeezing

is −1.For the COSTS, by using Eqs. (1), (3), (7), and (9) we derive

[see Appendix B]

ha†i ¼ Trfρa†g ¼ 0 ð12Þ

ha†2i ¼ 6AðBþ r2Þ þ 2ð8A2 þ B2 þ BÞrtBþ r2 þ 4Atr

; ð13Þ

ha†ai ¼ ð3Bþ 1Þðr2 þ 4trAÞ þ 4A2 þ 2B2

Bþ r2 þ 4Atr; ð14Þ

and the degree of squeezing is

S1 ≡ −26AðBþ r2Þ þ 2ð8A2 þ B2 þ BÞrt

Bþ r2 þ 4Atr

þ 2ð3Bþ 1Þðr2 þ 4trAÞ þ 4A2 þ 2B2

Bþ r2 þ 4Atr; ð15Þ

where A≡ ½ð2�nþ 1Þ sinh 2κ�=4 and B≡ �n cosh 2κ þ sinh2 κ. Inparticular, when �n ¼ 0, e−β → 0, ρth ¼ ð1 − e−βÞe−βa†a →

∶e−a†a∶ ¼ j0ih0j, ρs becomes the case of a squeezed vacuum

state, in this case A ¼ ðsinh 2κÞ=4, B ¼ sinh2 κ, which leadsto ha†2i ¼ ð3=2Þ sinh 2κ and ha†ai ¼ 3 sinh2 κ þ 1; thenEq. (15) reduces to

S�n¼0 ¼ 3e−2κ − 1; ð16Þ

which indicates that the degree of squeezing of the coherentoperated squeezed vacuum state does not change with r. Thisresult seems interesting, but not surprising. Actually, when�n ¼ 0, ρs ¼ SðκÞj0ih0jS†ðκÞ, due to aSðκÞj0i ¼ sinh κSðκÞj1i,a†SðκÞj0i ¼ cosh κSðκÞj1i; then from Eq. (1) we have ρ ¼SðκÞj1ih1jS−1ðκÞ (called a squeezed single-photon state).Therefore, applying coherent operation taþ ra† to thesqueezed vacuum state has the same impact as adding a singlephoton to (or subtracting a single photon from) it [41,42].

2150 J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011 Wang et al.

To compare S0 with S1, we draw two-dimensional graphs ofthe degree of squeezing using Eqs. (11) and (15) for differentparameters κ, �n, and r. Figure 1 shows that the value of S1 isalways larger than S0 for any r, which implies that the degreeof squeezing of the COSTS becomes weaker. Figure 1(a)shows that when �n ¼ 0, S1 converges for different values ofr, which represents Eq. (16). Figure 1(b) represents the �n ¼0:1 case, which shows that only within the range ½−1; 0Þ andfor small values of �n, S1 increases with r. We further find thatS1 increases with �n and becomes positive as �n is large enoughfor any values of κ and r. In particular, when κ ¼ 0, so thatA ¼ 0, B ¼ �n, Eq. (15) reduces to

Sκ¼0 ¼ 2ð3�nþ 1Þr2 − 2ð�n2 þ �nÞ þ 2�n2

�nþ r2; ð17Þ

which is always positive. That is to say, the coherent operationacting on a thermal state does not yield squeezing at all [26].

We now examine the sub-Poissonian statistics of theCOSTS in terms of the Mandel Q parameter, which is definedby

Q ¼ ha†2a2i − ha†ai2ha†ai : ð18Þ

Then, using Eqs. (1), (3), (7), and (9), after doing lengthybut straightforward calculation, we derive [see Appendix B]

Nha†2a2i ¼ NTrfρa†2a2g ¼ t2Bð6B2 þ 36A2Þþ r2ð36A2Bþ 20A2 þ 6B3 þ 10B2 þ 4BÞþ 24trAð2A2 þ 2B2 þ BÞ: ð19Þ

Substituting Eqs. (14) and (19) into Eq. (18), we obtain

Q¼ð20A2þ10B2þ4BÞr2þ24Að2A2þ2B2þBÞrtþ36A2Bþ6B3

ð3Bþ1Þðr2þ4AtrÞþ4A2þ2B2

−ð3Bþ1Þðr2þ4trAÞþ4A2þ2B2

r2þBþ4Atr: ð20Þ

When κ ¼ 0, so that A ¼ 0, B ¼ �n, then we have

Qκ¼0 ¼2�n2ð�nþ r2Þ2 − r4ð1þ �nÞ2ð�nþr2Þð2�n2 þ 3�nr2 þ r2Þ ; ð21Þ

which is just the Mandel Q parameter for the case of coherentoperation acting on a thermal state, so sub-Poissonian statis-tics emerge only when r2 >

ffiffiffi2

p�n2=½1 − ð ffiffiffi

2p

− 1Þ�n�. In particu-lar, when �n ¼ 0, ha†2a2i ¼ ð9=4Þ sinh2 2κ þ 6 sinh4 κ, Eq. (20)reduces to

Q�n¼0 ¼3 cosh 4κ − 6 cosh 2κ − 1

6 cosh 2κ − 2; ð22Þ

which indicates that the Mandel Q parameter for the coherentoperated squeezed vacuum state is the same as that of thesingle-photon-subtracted or added squeezed vacuum state.

Using Eq. (20), for the COSTS, the variation of theMandel Q parameter with r for different κ and �n is shownin Fig. 2. Figure 2(a) shows that for small values of both κand �n, the value of Q monotonically decreases with r andquickly becomes negative, which implies sub-Poissonian sta-tistics. For large �n, the variation of the Mandel Q parameterwith r is shown in Fig. 2(b), which is different from the caseof small �n.

4. WIGNER FUNCTIONS ANDNONCLASSICALITYThe Wigner function, first introduced by Wigner in 1932 [43],has now become a very popular tool to study the nonclassicalproperties of quantum states. The presence of the negativity ofthe Wigner function of an optical field is a signature ofnonclassicality. The Wigner function of ρ in the coherent staterepresentation jzi is expressed as [44]

WðαÞ ¼ 2e2jαj2

π

Zd2zπ h−zjρjzi exp½−2ðzα� − z�αÞ�; ð23Þ

where α ¼ xþ iy. To simplify the calculation, we decomposeρ of the COSTS into three components, i.e., the single-photonsubtraction STS aρsa†, the single-photon addition STS a†ρsa,and the “off-diagonal” components aρsa and a†ρsa†. Substitut-ing Eq. (1) into Eq. (23), and using the completeness relationof the coherent state and Eq. (7), we obtain the Wignerfunction as [see Appendix C]

0.0 0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

1.0

1.5

2.0 (b)(a)0n 0.1n

S S1S

0S

0S

Fig. 1. (Color online) Degree of quadrature squeezing S for the STS and the COSTS. (a) average photon number �n ¼ 0; bottom line, S0; top line, S1.(b) �n ¼ 0:1, except S0; other lines denote S1 for different values of r (from bottom to top r ¼ 0, 0.1, 0.3, 0.5, 0.8, 1).


WðαÞ ¼ W0ðαÞNð2�nþ 1Þ

�jtj2

�j2α�ð�n − sinh2 κÞ þ α sinh 2κj22 �nþ1

þ �n − sinh2 κ�

þ jrj2�j2α�ðnþ cosh2 κÞ − α sinh 2κj2

ð2�nþ 1Þ2 − �n − cosh2 κ�

þ ðr�tþ t�rÞN

�−ðα2 þ α�2Þ sinh2 2κ þ jαj2 sinh 4κ

2 �nþ1

−sinh 2κ

2

�þ ðr�tα2 þ t�rα�2Þ 4 �nð�nþ 1Þ

Nð2�nþ 1Þ�; ð24Þ

where W0ðαÞ is the Wigner function of STS

W0ðαÞ ¼2

πð2�nþ 1Þ exp�sinh 2κ2nþ 1

ðα�2 þ α2Þ − 2 cosh 2κ2nþ 1

jαj2�:

ð25Þ

In particular, when κ ¼ 0, Eq. (24) reduces to

W κ¼0ðαÞ ¼ð1þ 2�nÞWthðαÞð1þ �nÞðjrj2 þ �nÞ

��njtj2

1þ 2�n

−ð1þ 2�nÞjrj2 þ �n2

ð1þ 2�nÞ2�1 −

4ð1þ �nÞjαj21þ 2�n

�

þ 4�nð1þ �nÞ2ð1þ 2�nÞ3 ðtr

�α2 þ t�rα�2Þ�; ð26Þ

where W thðαÞ ¼ 2πð2�nþ1Þ e

−2jαj22nþ1 is the Wigner function of a ther-

mal state. When �n ¼ 0, Eq. (24) reduces to theWigner functionof a single-photon-subtracted (or added) squeezed vacuumstate [19],

W �n¼0ðαÞ ¼ W 00ðαÞð−1þ 4jα� sinh κ − α cosh κj2Þ; ð27Þ

where W 00ðαÞ ¼ 2

π exp½ðα�2 þ α2Þ sinh 2κ − 2jαj2 cosh 2κ�, theWigner function of squeezed vacuum state.

From Eq. (24), we can infer that the negative region ofWigner Wðα ¼ xþ iyÞ < 0 appears under the condition

C1x2 þ C2y

2 < C3; ð28Þ

where

C1 ¼ 4e−2κr2 þ 24 �nð�nþ 1Þ − e−4κ þ 1

2 �nþ1rt

þ ½sinh 2κ þ 2ð�n − sinh2 κÞ�22 �nþ1

;

C2 ¼ 4e2κr2 − 24 �nð�nþ 1Þ − e4κ þ 1

2 �nþ1rt

þ ½sinh 2κ − 2ð�n − sinh2 κÞ�22 �nþ1

;

C3 ¼ ð2�nþ 1Þr2 þ rt sinh 2κ − ð�n − sinh2 κÞ:Equation (28) clearly indicates that the contour line of thenegative region of Wigner functions is a set of ellipses, with

the size of the negative area given by S ¼ πC3ffiffiffiffiffiffiffiffiC1C2

p . We can prove

that C1 and C2 are always positive; therefore, the negative re-gion emerges under the condition C3 > 0. We can also inferthat the negative area increases with r for small κ. However,the negative area does not always increase with r for large κ.Obviously, when sinh2 κ > �n, the Wigner function always hasnegative values for any r. When κ ¼ 0, the negativity region

emerges only under the condition r >ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�n=ð1þ 2�nÞp

[26].Using Eq. (24), the Wigner functions are depicted in

Figs. 3–5 for different values of κ and r. There are some partialnegative regions of Wigner functions as shown in Figs. 3 and 4,which is the evidence of the nonclassicality of the COSTS.

Figure 3 shows that both negative dip and the area of thenegative region of the Wigner function increase with κ. TheWigner functions obviously have two positive peaks and onenegative peak. Meanwhile, the profiles of theWigner functionsquickly become narrow in momentum quadrature, but spreadwidely in position quadrature, which implies that the degree ofsqueezing is becoming stronger when κ increases.

Figure 4, for fixed both �n and κ, shows that both negativedip and the area of the negative region of the Wigner functionincrease with r. This indicates that the photon addition com-ponent a†ρsa greatly affects the negativity of the Wigner func-tion, which provides supporting evidence to the results ofRef. [20]. The contribution to the Wigner function from the“off-diagonal” component aρsaþ a†ρsa† is shown in Fig. 5for different κ.

5. DECOHERENCE INAMPLITUDE-DAMPING CHANNELWhen the COSTS evolves in the amplitude-damping channel,its density matrix obeys the master equation

0.0 0.2 0.4 0.6 0.8 1.00.8

0.6

0.4

0.2

0.0

0.2

0.0 0.2 0.4 0.6 0.8 1.0

4.1

4.2

4.3

4.4

4.5

4.60.1n 1.0n(a) (b)

QQ0.3

0.62

0.2 0.61

0.1 0.60

r r

Fig. 2. (Color online) Q parameter of the COSTS changes with r for different values of both parameters �n and κ.


dρdt

¼ γð2aρa† − a†aρ − ρa†aÞ; ð29Þ

where γ denotes the damping coefficient. The time evolutionof the Wigner function is given by the integration equation [45]

Wðz; tÞ ¼ 2T

Zd2απ Wðα; 0Þ exp

�−2

jz − αe−γtj2T

�; ð30Þ

where Wðα; 0Þ is the Wigner function of the initial stateand T ¼ 1 − e−2γt.

Substituting Eqs. (24) and (25) into Eq. (30), we obtain [seeAppendix D]

Wðz; tÞ ¼ W0ðz; tÞFðz; tÞ; ð31Þ

whereW0ðz; tÞ is the Wigner function of the STS evolved in theamplitude-damping channel

W0ðz; tÞ ¼2=ð2�nþ 1ÞπT

ffiffiffiffiG

p exp

�Δ1jzj2 þ

g2g23

4Gðz�2 þ z2Þ

�; ð32Þ

and

NFðz; tÞ ¼ jtj2ðχ þ jωj2Þ þ jrj2ðχ 0 þ jω0j2Þ

þ ðr�tþ t�rÞ�Δ2 þ 2μυ − g32

G− μ2 − μ�2

þ 4 �nð�nþ 1Þð2�nþ 1Þ2

�g2

2Gðrt� þ r�tÞ þ rt�ξ2 þ r�tξ�2

�; ð33Þ

where

g0 ¼cosh 2κ2 �nþ1

; g1 ¼�n − sinh2 κ2 �nþ1

; g2 ¼sinh 2κ2 �nþ1

;

g3 ¼2e−γt

T; g01 ¼ 1 − g1; ð34Þ

as well as

Fig. 3. (Color online) Wigner function of COSTS for r ¼ 0:3 and �n ¼ 0:1 with (a) κ ¼ 0, (b) κ ¼ 0:2, and (c) κ ¼ 0:6. (d), (e), and (f) are contourlines with interval 0.05, corresponding to (a), (b), and (c), respectively, where only the negative regions are plotted.


δ ¼ 2g0 þ g3e−γt; G ¼ δ2

4− g22; Δ1 ¼

−2T

þ δg234G

; Δ2 ¼ −g2

2þ δg0g2

2G; μ ¼ δg2g3zþ 2g22g3z

�

4G;

χ ¼ δð4g21 þ g22Þ þ 8g1g224G

þ g1; χ 0 ¼ δð4g021 þ g22Þ − 8g01g22

4G− g01; ω ¼ δg3ð2g1zþ g2z

�Þ þ g2g3ð4g1z� þ 2g2zÞ4G

;

ω0 ¼ δg3ð2g01z − g2z�Þ þ g2g3ð4g01z� − 2g2zÞ

4G; υ ¼ g0g3ðδz� þ 2g2zÞ

4G; ξ ¼ g3ðδz� þ 2g2zÞ

4G: ð35Þ

Fig. 4. (Color online) Wigner function of COSTS for squeezing parameter κ ¼ 0:6 and �n ¼ 0:1with (a) r ¼ 0:1, (b) r ¼ 0:5, and (c) r ¼ 1:0. (d), (e)and (f) are contour lines with interval 0.05, corresponding to (a), (b), and (c), respectively, where only the negative regions are plotted.

Fig. 5. (Color online) Contribution to Wigner function of COSTS from the “off-diagonal” components aρsaþ a†ρsa† for r ¼ 0:3 and �n ¼ 0:1 with(a) κ ¼ 0 and (b) κ ¼ 0:6.


Actually, one can infer from Eq. (31) that when r ¼ 0,Eq. (31) reduces to the Wigner function of the single-photon-subtraction STS in the amplitude-damping channel.When r ¼ 1, Eq. (31) reduces to the Wigner function of thesingle-photon-addition STS in the amplitude-damping channel.

From Eq. (31) one can see that when the factor Fðz; tÞ < 0,the Wigner function Wðz; tÞ has its negative distribution inphase space. At the center, z ¼ 0, Wð0; tÞ always has negativevalues when

jtj2χ þ jrj2χ 0 þ ðr�tþ t�rÞ�Δ2 −

g32G

þ 4 �nð�nþ 1Þg22ð2�nþ 1Þ2G

�< 0: ð36Þ

This inequality is hard to solve, however, when r ¼ 0; thesolution is χ < 0, leading to the following condition:

γt < γtc ¼12ln

�1 −

ð2�nþ 1Þð�n − sinh2 κÞ�n cosh 2κ þ sinh2 κ

�; ð37Þ

which indicates that the Wigner function is always positive inphase space when γt exceeds the threshold value γtc.

Using Eq. (31), we present the time evolution of theWigner function at different time scales in Fig. 6, whichexhibits how the negative region of the Wigner functiongradually diminishes as γt increases. When γt → ∞,δ → ð2 cosh 2κÞ=ð2�nþ 1Þ, ω ¼ ω0

→ 0, μ ¼ υ → 0, ξ → 0, G →

1=ð2�nþ 1Þ2, Δ1 → −2, Δ2 − g32=G → ðsinh 2κÞ=2ð2�nþ 1Þ, χ →

�n cosh 2κ þ sinh2 κ, and χ0 → �n cosh 2κ þ cosh2 κ; notingEq. (9), we have

Wðz;∞Þ ¼ 2π exp½−2jzj

2�; ð38Þ

which corresponds to vacuum states. This result is the sameas that of Ref. [19]. It implies that the system reduces to avacuum state after a long time.

The changes of Wigner functions with r for fixed both κ andthe decay time γt are shown in Fig. 7. We find that the negativedip of the Wigner function increases with r in the amplitude-

damping channel. Therefore, compared with the single-photon-subtraction STS [15,19], by modulating the proportionr, the threshold value of the decay time may be enlarged bythe coherent operation.

Fig. 6. (Color online) Wigner function of COSTS in amplitude-damping channel for κ ¼ 0:3, r ¼ 0:3, n ¼ 0:1, and different γt: (a) γt ¼ 0:05,(b) γt ¼ 0:2, (c) γt ¼ 0:5, and (d) γt ¼ 0:7.

Fig. 7. (Color online) Wigner function of COSTS in amplitude-damping channel for κ ¼ 0:6, γt ¼ 0:3, �n ¼ 0:1, and different r:(a) r ¼ 0:0, (b) r ¼ 0:3, and (c) r ¼ 1.


6. CONCLUSIONIn summary, we have studied the nonclassicality and decoher-ence of the COSTS when the coherent operation taþ ra† actson an STS. Based on the normal ordering form of the densityoperator of STS, the quadrature squeezing and Mandel Q

parameter are derived analytically. Compared with the STS,the degree of squeezing of the COSTS becomes weaker anddecreases when r increases. For a small squeezing amount,κ, the Mandel Q parameter monotonically decreases with r

and quickly becomes negative, which implies sub-Poissonianstatistics. On the other hand, when κ is large enough, Q isalways positive for any r. When the thermal photon number�n ¼ 0, in the case of a pure squeezed state being an initialstate, the COSTS reduces to the squeezed single-photon state.

Both the negative dip and negative area of the Wigner func-tion of a COSTS increase with r. In addition, the Wigner func-tion of a COSTS is always negative when sinh2 κ > �n for any r.We also found that the threshold value of the decay time cor-responding to the transition of Wigner functions from partiallynegative to completely positive can be enlarged by the coher-ent operation, which may be a practically important result tothe quantum information. All these results indicate that thenonclassicality of the STS is sensitive to the coherent opera-tion of photon addition and subtraction.

APPENDIX A: DERIVATION OFEQUATION (8)In order to obtain the normalization constant N of ρ, wedecompose it into four components as follows:

N ¼ Trfjtj2aρsa†g þ Trfjrj2a†ρsag þ Trfr�taρsagþ Trfrt�a†ρsa†g: ðA1Þ

Using the completeness relation of the coherent state,Eq. (7) and ρs in Eq. (3), as well as the overlap of the coherentstate,

hz1jz2i ¼ exp

�−12jz1j2 −

12jz2j2 þ z�1z2

�; ðA2Þ

we have

Trfaρsa†g ¼ 1τ1τ2

∂2

∂s∂f

Zdz1π exp

�−τ21 þ τ222τ21τ22

jz1j2

þ�1 −

τ21 þ τ222τ21τ22

�f z�1 þ

�sþ f

τ21 − τ222τ21τ22

�

× z1τ21 − τ22þ4τ21τ22

ðz21 þ z�21 Þ þ τ21 − τ224τ21τ22

f 2�s¼f¼0

¼ ∂2

∂s∂fexp

�τ21 þ τ22 − 22

sf þ τ21 − τ224

ðf 2 þ s2Þ�s¼f¼0

¼ τ21 þ τ22 − 22

ðA3Þ

and

Trfa†ρsag ¼ ∂2

∂s∂fexp

�τ21 þ τ222

sf þ τ21 − τ224


¼ ðτ21 þ τ22Þ2

; ðA4Þ

as well as

Trfr�taρsaþ rt�a†ρsa†g ¼ τ21 − τ222

ðr�tþ rt�Þ: ðA5Þ

Therefore, substituting Eqs. (A3)–(A5) into (A1), we obtainEq. (8).

APPENDIX B: DERIVATION OFEQUATIONS (13), (14), AND (19)When the coherent operation acts on the STS, the averagevalue of a†2 in the COSTS is

ha†2i ¼ Trfρa†2g: ðB1Þ

Here we consider t and r as being real positive numbers.Substituting Eq. (1) into Eq. (B1) and using an approachsimilar to the calculation of Eq. (A3), we obtain

Trfaρsa†a†2g ¼ ∂4

∂s∂f 3exp

�τ21 þ τ22 − 22

sf

þ τ21 − τ224


¼ 6AB; ðB2Þ

and

Trfa†ρsaa†2g ¼ ∂4

∂s3∂fexp

�ðτ21 þ τ22Þ2

f sþ τ21 − τ224


¼ 6AðBþ 1Þ; ðB3Þ

where we have set A ¼ ðτ21 − τ22Þ=4 and B ¼ ðτ21 þ τ22 − 2Þ=2. Byusing a similar method to the calculation of Eq. (A4), one canderive

Trfaρsaa†2g ¼ Trfa†2ρsa2g − 2Trfa†ρsag ¼ 4A2 þ 2B2 þ 2B;

ðB4Þ

Trfa†ρsa†a†2g ¼ 12A2: ðB5Þ

Substituting Eqs. (B2)–(B5) into Eq. (B1), we obtainEq. (13). Similarly, we can prove

ha†i ¼ Trfρa†g ¼ 0: ðB6Þ

For the average photon number in the COSTS, we evaluate

ha†ai ¼ 1NTrft2a2ρsa†2 þ r2ða†2ρsa2 − a†ρsaÞ

þ rtða3ρsa† þ a2ρs þ aρsa†3 þ ρsa†2Þg: ðB7Þ


Then, using the approach for deriving Eqs. (A3) and (A4),we obtain Eq. (14). For the average of a†2a2 in the COSTS, wehave

ha†2a2i ¼ 1NTrft2a3ρsa†3 þ r2a†ρsaa†2a2

þ rtða3ρsaa†2 þ a†ρsa†a†2a2Þg: ðB8Þ

Adopting the approach used above, we obtain Eq. (19).

APPENDIX C: DERIVATION OF WIGNERFUNCTION OF EQUATION (24)For simplifying the calculation, one can decompose ρ of theCOSTS into three components, i.e.,

W−ðαÞ ¼e2jαj2 jtj2Nπ

Zd2zπ h−zjaρsa†jzie−2ðzα�−z�αÞ; ðC1Þ

WþðαÞ ¼e2jαj2 jrj2Nπ

Zd2zπ h−zja†ρsajzie−2ðzα�−z�αÞ; ðC2Þ

W 0ðαÞ ¼ e2jαj2

Nπ

Zd2zπ h−zjðtr�ρsa2 þ t�rρsa†2Þjzie−2ðzα�−z�αÞ;

ðC3Þ

Using the completeness relation of coherent state and ρs inEq. (3), as well as Eq. (A2), Eq. (C1) can be written as

W−ðαÞ ¼e2jαj2 jtj2Nπτ1τ2

∂2

∂f ∂s

Zd2zπ exp½−jzj2 − 2ðzα� − z�αÞ�

×Z

d2z1π exp

�−jz1j2 − ðz� − sÞz1 þ

τ21 − τ224τ21τ22

z�21

�

×Z

d2z2π exp

�−jz2j2 þ z�1z2 −

τ21 þ τ222τ21τ22

z�1z2

þ ðzþ f Þz�2 −z22

4τ21þ z22

4τ22

�s¼f¼0

: ðC4Þ

Then, using Eq. (7), we obtain

W−ðαÞ ¼jtj2W0ðαÞNð2nþ 1Þ

�jα sinh 2κ þ 2α�ðn − sinh2 κÞj22nþ 1

þ n

− sinh2 κ�; ðC5Þ

where W0ðαÞ is the Wigner function of STS. In the same way,we obtain

WþðαÞ ¼jrj2W0ðαÞNð2�nþ 1Þ

�j2α�ðnþ cosh2 κÞ − α sinh 2κj22 �nþ1

− �n

− cosh2 κ�: ðC6Þ

The contribution of the “off-diagonal” component Eq. (C3)is

W 0ðαÞ ¼ ðr�tþ t�rÞW0ðαÞNð2�nþ 1Þ

�−sinh 2κ

2

þ −ðα2 þ α�2Þ sinh2 2κ þ jαj2 sinh 4κð2�nþ 1Þ

�

þ ðr�tα2 þ t�rα�2Þ 4 �nð�nþ 1ÞNð2�nþ 1Þ2 W0ðα; α�Þ: ðC7Þ

Thus, the Wigner function of the COSTS is obtained.

APPENDIX D: DERIVATION OF WIGNERFUNCTION OF EQUATION (31)In order to obtain Eq. (31), we can substitute Eqs. (24) and(C5)–(C7) into Eq. (30), successively. First, substitutingEq. (C5) into Eq. (30), we obtain

W−ðz; tÞ ¼2jtj2T N

Zd2απ W0ðαÞ exp

�−2

jz − αe−γtj2T

�

×

��n − sinh2 κ2 �nþ1

þ j2α�ð�n − sinh2 κÞ þ α sinh 2κj2ð2�nþ 1Þ2

�

¼ 4jtj2T Nπð2�nþ 1Þ exp

�−2jzj2T

�∂2

∂f ∂s

�exp½f sg1�

×Z

d2απ exp½−ð2g0 þ g3e

−γtÞjαj2

þ ð2f g1 þ sg2 þ g3z�Þα

þ ð2sg1 þ f g2 þ g3zÞα� þ g2ðα�2 þ α2Þ��

f¼s¼0: ðD1Þ

Then, using Eq. (7), we obtain

W−ðz; tÞ ¼2jtj2ðχ þ jωj2Þ

NπTffiffiffiffiG

p ð2�nþ 1Þ exp�Δ1jzj2 þ

g2g23

4Gðz�2 þ z2Þ

�:

ðD2Þ

Then by the same approach as used in deriving Eq. (D1), sub-stituting Eqs. (C6) and (C7) into Eq. (30), after doing lengthybut straightforward calculation, we can finally obtain Eq. (31).

ACKNOWLEDGMENTSWe sincerely thank Prof. Hyunchul Nha for his enlighteningdiscussions. We also thank the referees for helpful sugges-tions. This work is supported by the National Natural ScienceFoundation of China (NSFC), grants 10874174 and 11047133,and the Shandong Provincial Natural Science Foundation inChina (grant ZR2010AQ024).

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