se to

t turns out

Physics Letters A 327 (2004) 290–295

www.elsevier.com/locate/pla

Spectral properties of the squeeze operator

Dariusz Chruscinski

Institute of Physics, Nicolaus Copernicus University, ul. Grudziadzka 5/7, 87-100 Torun, Poland

Received 24 March 2004; accepted 24 May 2004

Available online 2 June 2004

Communicated by P.R. Holland

Abstract

We show that a single-mode squeeze operatorS(z) being an unitary operator with a purely continuous spectrum gives ria family of discrete real generalized eigenvalues. These eigenvalues are closely related to the spectral properties ofS(z) and thecorresponding generalized eigenvectors may be interpreted as resonant states well known in the scattering theory. Ithat these states entirely characterize the action ofS(z). This result is then generalized toN-mode squeezing. 2004 Elsevier B.V. All rights reserved.

PACS: 42.50.-p; 02.30.Tb; 42.50.Dv

Keywords: Squeezed states; Spectral analysis

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1. Introduction

Squeezed states play a prominent role in the mern quantum optics, see, e.g.,[1–3]. Mathematicalproperties of these states were investigated in aries of papers in the 1970s and early 1980s, see,[4–8]. However, they were already discovered by Kenard [9] (see the historical reviews by Nieto[10]).Squeezed states are quantumstates for which no classical analog exists. Recently, they have drown a gdeal of interest in connection with quantum informtion theory. Squeezed states of light were successteleported in the experiment reported by the groupFurusawa[11] (see also[12]).

E-mail address: darch@phys.uni.torun.pl (D. Chruscinski).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.05.046

,

In the present Letter we analyze the spectral prerties of single-modeS(z) andN -mode squeeze opeators. Clearly, squeeze operators are unitary and htheir spectra define a subset of complex numbersmodulus one. It is easy to show that the spectrumthe squeeze operator is purely continuous and cothe entire unit circle on the complex plane. Howevit is not the whole story. Actually, it is easy to nothatS(z) displays two families of discrete real eigevalues. Clearly, these eigenvalues are not proper,is, the corresponding eigenvectors do not belong tocorresponding Hilbert space of square integrable futions. One of these families was reported in a seriepapers by Jannussis et al.[13,14]. However, the inter-pretation of this result was not clear (it was criticizby Ma et al.[15] who stressed that proper eigenvaluof the squeeze operator do not exist).

.

D. Chruscinski / Physics Letters A 327 (2004) 290–295 291

isen-

vec-own

oflic

, co-

reteThisez-

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d-ns

. Itmt itdard

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In the present Letter we are going to clarify thproblem. In particular we show that the above mtioned discrete eigenvalues ofS(z) are closely relatedto its spectral properties. The corresponding eigentors may be interpreted as resonant states well knin the scattering theory[16,17]. Actually, generalizedeigenvectors ofS(z) correspond to resonant statesso-called inverted oscillator (or equivalently parabopotential barrier)[18] (see also[19–24]). We show thatrestricting oneself to a suitable class of states (e.g.herent states do belong to this class) the action ofS(z)

may be entirely characterized in terms of these disceigenvalues and the corresponding eigenvectors.observation is then generalized to two-mode squeing and finally toN -mode squeezing.

2. Single-mode squeezing

A single-mode squeeze operator is defined by

(2.1)S(z) = exp

(1

2

[za†2 − z∗a2]),

where z is a complex number anda (a†) is thephoton annihilation (creation) operator which obeystandard commutation relation[a, a†] = 1. Clearly,S(z) may be represented asS(z) = exp(iH(z)), with

(2.2)H(z) = 1

2i

(za†2 − z∗a2).

Now, to investigate spectral properties ofH(z) let usconsider a unitarily equivalent operatorR†(ϕ)H(z) ×R(ϕ), whereR(ϕ) is a single-mode rotation[25–27]

(2.3)R(ϕ) = exp(iϕa†a

).

One has

(2.4)R†(ϕ)H(z)R(ϕ) = H(ze−2iϕ

),

and hence, forϕ = θ/2, wherez = reiθ , it shows thatH(z) is unitarily equivalent toH(r). Introducing twoquadraturesx andp via

(2.5)a = x + ip√2

, a† = x − ip√2

,

one finds the following formula forH(r):

(2.6)H(r) = − r

2(xp + px).

Spectral properties ofH(r) were recently investigatein [28] (see also[29]) in connection with quantum dissipation. Note, that the classical Hamilton equatioimplied by the HamiltonianH = −rxp:

(2.7)x = −rx, p = rp,

describe the damping ofx and pumping ofp. Thisis a classical picture of the squeezing processturns out thatH(r) has purely continuous spectrucovering the whole real line. Hence, it is clear thadoes not have any proper eigenvalue. Using stanSchrödinger representation forx andp = −id/dx, thecorresponding eigen-problemH(r)ψ = Eψ may berewritten as follows

(2.8)xd

dxψ(x) = −

(iE

r+ 1

2

)ψ(x).

Note, thatH(r) is parity invariant and hence eacgeneralized eigenvalueE ∈ R is doubly degeneratedTherefore, two independent solutions of(2.8) aregiven by

(2.9)ψE± (x) = 1√2πr

x−(iE/r+1/2)± ,

wherexλ± are distributions defined as follows[30] (seealso[31]):

xλ+ :={

xλ x � 0,

0 x < 0,

(2.10)xλ− :={

0 x � 0,

|x|λ x < 0,

with λ ∈ C (basic properties ofxλ± are collectedin [28]). These generalized eigenvectorsψE± are com-plete

(2.11)∫

ψE± (x)ψE± (x ′) dE = δ(x − x ′),

andδ-normalized

(2.12)∫

ψE1± (x)ψ

E2± (x) dx = δ(E1 − E2).

Hence they give rise to the following spectral resotion of H(r):

(2.13)H(r) =∑±

∫E

∣∣ψE±⟩⟨ψE±

∣∣dE,

292 D. Chruscinski / Physics Letters A 327 (2004) 290–295

eze

-of

f

mx

al.,

It

d,

he-sh

allede

ow-

tor

in-en--

or

and the corresponding spectral resolution of squeoperatorS(r) immediately follows

(2.14)S(r) =∑±

∫eiE

∣∣ψE±⟩⟨ψE±

∣∣dE.

Now, let us observe thatFH = −HF , whereF de-notes the Fourier transformation. Hence, ifH(r)ψE =EψE , then H(r)F [ψ−E] = EF [ψ−E]. Therefore,the family F [ψ−E± ] defines another system of complete andδ-normalized generalized eigenvectorsH(r). Note that the action ofS(r) is defined by

(2.15)S(r)ψ(x) = e−r/2ψ(e−rx

),

and its Fourier transform

(2.16)F[S(r)ψ

](p) = er/2F [ψ](erp

),

due toFS(r) = S(−r)F , that is, if the fluctuations op are reduced then the fluctuation ofx are amplifiedand vice versa.

3. Discrete real eigenvalues of S(r)

Surprisingly, apart from the continuous spectruH(r) gives rise to the following families of complediscrete eigenvalues[28]

(3.1)H(r)f ±n = ±Enf

±n ,

where

(3.2)En = ir

(n + 1

2

),

and

(3.3)f −n (x) = (−1)n√

n! δ(n)(x), f +n (x) = xn

√n! .

Interestingly they satisfy the following properties:

(3.4)

∞∫−∞

f +n (x)f −

m (x) dx = δnm,

and

(3.5)∞∑

n=0

f +n (x)f −

n (x ′) = δ(x − x ′).

It implies that

(3.6)S(r)f ±n = e±iEnf ±

n ,

which shows thatS(r) displays two families of purelyreal generalized eigenvalues

(3.7)s±n = exp

[±r

(n + 1

2

)].

A family s+n was already derived by Jannussis et

see, e.g., formula (2.3) in[14], but they overlooked thesecond ones−

n . How to interpret these eigenvalues?turns out that one recoversEn and f ±

n by studyinga continuation of generalized eigenvectorsψE± andF [ψ−E± ] into the energy complex planeE ∈ C. BothψE andF [ψ−E ] display singular behavior whenE iscomplex:ψE± has simple poles atE = −En, whereasF [ψ−E± ] has simple poles atE = +En, with En

defined in(3.2). Moreover, their residues corresponup to numerical factors, to the eigenvectorsf ±

n :

(3.8)Res(ψE± (x);−En

) ∼ f −n ,

and

(3.9)Res(F

[ψ−E± (x)

];+En

) ∼ f +n .

Such eigenvectors are well known in scattering tory as resonant states, see, e.g.,[16] and referencetherein. In the so-called rigged Hilbert space approacto quantum mechanics these states are also cGamov vectors[17]. To see the connection with thscattering theory let us observe that under the folling canonical transformation:

(3.10)x = rQ − P√2r

, p = rQ + P√2r

,

H(r) transforms into the unitarily equivalent opera[18]

(3.11)H(r) → Hio = 1

2

(P 2 − r2Q2),

which represents the Hamiltonian of the so-calledverted or reversed oscillator (or equivalently a pottial barrier ‘−r2Q2/2’) and it was studied by several authors in various contexts[19–24]. An invertedoscillatorHio corresponds to the harmonic oscillatwith a purely imaginary frequencyω = ±ir and hencethe harmonic oscillator spectrum ‘ω(n+1/2)’ implies‘±ir(n + 1/2)’ as generalized eigenvalues ofHio.

D. Chruscinski / Physics Letters A 327 (2004) 290–295 293

r

ble

r

tity

ze

onis

f

to

a-l

ix.

4. A new representation of S(r)

Interestingly, the action ofS(r) may be entirelycharacterized in terms off ±

n ands±n . Indeed, conside

a spaceD of smooth functionsψ = ψ(x) withcompact supports, i.e.,ψ(x) = 0 for |x| > a for somepositivea (depending upon chosenψ), see, e.g.,[32].Clearly, D defines a subspace of square integrafunctions L2(R). Moreover, letZ = F [D], that is,ψ ∈ Z if ψ = F [φ] for someφ ∈ D. It turns out[32]thatD andZ are isomorphic andD ∩ Z = ∅. Now,any functionφ from Z may be expanded into Tayloseries and hence

(4.1)φ(x) =∞∑

n=0

φ(n)(0)

n! xn =∞∑

n=0

f +n (x)

⟨f −

n

∣∣φ⟩.

On the other hand, for anyφ ∈D, its Fourier transformF [φ] ∈ Z, and

φ(x) = 1√2π

∫eikxF [φ](k) dk

= 1√2π

∫eikx

∞∑n=0

F [φ](n)(0)

n! kn dk

=∞∑

n=0

F[f +

n

](x)

⟨f −

n

∣∣F [φ]⟩

(4.2)=∞∑

n=0

f −n (x)

⟨f +

n

∣∣φ⟩.

Hence, we have two decompositions of the idenoperator

(4.3)1 =∞∑

n=0

∣∣f +n

⟩⟨f −

n

∣∣ onZ,

and

(4.4)1 =∞∑

n=0

∣∣f −n

⟩⟨f +

n

∣∣ onD.

It implies the following representations of the squeeoperatorS(r):

(4.5)S(r) =∞∑

n=0

s−n

∣∣f +n

⟩⟨f −

n

∣∣ onZ,

and

(4.6)S(r) =∞∑

s+n

∣∣f −n

⟩⟨f +

n

∣∣ onD.

n=0

It should be stressed that the above formulae forS(r)

are not spectral decompositions and they valid onlyZ andD, respectively (its spectral decompositiongiven in formula(2.14)). Note, thatS(r) mapsZ intoD and using(4.5)one has

(4.7)S†(r) =∞∑

n=0

s−n

∣∣f −n

⟩⟨f +

n

∣∣ = S(−r) onD.

Conversely,S(r) represented by(4.6)mapsD into ZandS†(r) = S(−r) on Z. It shows that squeezing ox (p) corresponds to amplifying ofp (x). Clearly, ageneral quantum stateψ ∈ L2(R) belongs neither toDnor toZ. An example of quantum states belongingZ is a family of Glauber coherent states|α〉. Consider,e.g., a squeezed vacuumS(r)ψ0, where ψ0(x) =π−1/4e−x2/2. One has

(4.8)ψ0(x) = 1

π1/4

∞∑n=0

(−1)n√(2n)! f +

2n(x),

and

S(r)ψ0(x) = e−r/2

π1/4

∞∑n=0

(−e−2r )n√(2n)! f +

2n(x)

(4.9)= e−r/2

π1/4

∞∑n=0

(−1)n√(2n)!f

+2n

(e−rx

).

The similar formulae hold forS(r)|α〉.

5. Two-mode squeezing

Consider now a two-mode squeeze operator[25–27]

(5.1)S2(z) = exp(za

†1a

†2 − z∗a1a2

),

wherea†k andak are creation and annihilation oper

tors for two modesk = 1,2. Introducing 2-dimensionavectors

(5.2)aT = (a1, a2),(a†)T = (

a†1, a

†2

),

one finds

(5.3)S2(z) = exp(z(a†)T

σ1a† − z∗aTσ1a

),

whereσ1 stands for the corresponding Pauli matrUsing well-known relation

(5.4)eiπ/4σ2σ1e−iπ/4σ2 = σ3,

294 D. Chruscinski / Physics Letters A 327 (2004) 290–295

-

r-e

of)

e

d

s

nedera-eons

one obtains

eiπ/4σ2S2(z)e−iπ/4σ2

= exp(z(a†)T

σ3a† − z∗aTσ3a

)(5.5)= S(1)(z)S(2)(−z),

whereS(k) denotes a single-mode(k) squeeze operator. Now, sinceS(k)(z) is unitarily equivalent toS(k)(r)

a two-mode squeeze operatorS2(z) is unitarily equiv-alent toS(1)(r)S(2)(−r). Hence, the spectral propeties ofS2(z) easily follows. In particular defining thspaceD of smooth functionsψ = ψ(x1, x2) with com-pact supports andZ = F [D] one obtains the followingrepresentations:

(5.6)S(1)(r)S(2)(−r) =∞∑

nm=0

s−nm

∣∣f +nm

⟩⟨f −

nm

∣∣ onZ,

and

(5.7)S(1)(r)S(2)(−r) =∞∑

nm=0

s+nm

∣∣f −nm

⟩⟨f +

nm

∣∣ onD,

where

(5.8)f ±nm(x1, x2) = f ±

n (x1)f±m (x2),

and

(5.9)s±nm = e±r(n−m).

Jannussis et al.[14] claimed that the eigenvaluesS2(z) are given bye2(m−n), see, e.g., formula (5.7in [14]. Their result has the similar form ass−

nm butof course it is incorrect. Note that eigenvalues of[14]do not depend upon the squeezing parameterz as wasalready observed in[15].

6. N -mode squeezing

Following [33] one defines anN -mode squeezoperator

(6.1)SN (Z) = exp

(1

2

(a†)T

Za† − 1

2a†Z†a

),

whereZ is anN ×N symmetric (complex) matrix an

aT = (a1, a2, . . . , aN).

Defining anN -mode rotation operator

(6.2)RN(Φ) = exp(i(a†)T

Φa),

with Φ being anN × N Hermitian matrix, one show[27,33]

(6.3)R†N(Φ)SN (Z)RN(Φ) = SN

(e−iΦ Ze−iΦT)

.

Now, by a suitable choice ofΦ one obtains

(6.4)e−iΦ Ze−iΦT = ZD,

whereZD is a diagonal matrix, i.e.,(ZD)kl = zkδkl .Hence, anN -mode squeeze operatorSN (Z) is unitar-ily equivalent to

R†N(Φ)SN (Z)RN(Φ)

(6.5)= S(1)(z1)S(2)(z2) · · ·S(N)(zN),

and therefore its properties are entirely goverby the properties of the single-mode squeeze optor S(z). In particularSN(Z) gives rise to a discretfamily of generalized eigenvalues being combinatiof generalized eigenvalues ofS(k)(zk). Defining thecorresponding subspacesD andZ in the Hilbert spaceL2(RN) one easily finds

R†N(Φ)SN (Z)RN(Φ)

=∞∑

n1,...,nN =0

s−n1...nN

∣∣f +n1...nN

⟩⟨f −

n1...nN

∣∣ onZ,

(6.6)

and

R†N(Φ)SN (Z)RN(Φ)

=∞∑

n1,...,nN =0

s+n1...nN

∣∣f −n1...nN

⟩⟨f +

n1...nN

∣∣ onD,

(6.7)

where

(6.8)f ±n1...nN

(x1, . . . , xN) = f ±n1

(x1) · · ·f ±nN

(xN),

and

s±n1...nN

= exp

{±

[r1

(n1 + 1

2

)+ · · ·

(6.9)+ rN

(nN + 1

2

)]},

with rk = |zk|.

D. Chruscinski / Physics Letters A 327 (2004) 290–295 295

shh-3.

iv.

g,

07.

f6,

H.J.

.

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nder-

7

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5...

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Acknowledgements

This work was partially supported by the PoliMinistry of Scientific Research and Information Tecnology under the grant No. PBZ-MIN-008/P03/200

References

[1] D. Walls, Nature 306 (1983) 141.[2] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge Un

Press, Cambridge, 1997.[3] D.F. Walls, G.J. Milburn, Quantum Optics, Springer-Verla

Berlin, 1999.[4] D. Stoler, Phys. Rev. D 1 (1970) 3217;

D. Stoler, Phys. Rev. D 4 (1974) 1925.[5] H. Yuen, Phys. Rev. A 13 (1976) 2226.[6] J.N. Hollenhorst, Phys. Rev. D 19 (1979) 1669.[7] R. Fisher, M. Nieto, V. Sandberg, Phys. Rev. D 29 (1984) 11[8] D. Truax, Phys. Rev. D 31 (1985) 1988.[9] E.H. Kennard, Z. Phys. 44 (1927) 326.

[10] M.M. Nieto, in: G.T. Moore, M.O. Scully (Eds.), Frontiers oNonequilibrium Statistical Physics, Plenum, New York, 198p. 287;M.M. Nieto, quant-ph/9708012.

[11] A. Furusawa, J.L. Sorensen, S.L. Braunstein, C.A. Fuchs,Kimble, E.J. Polzik,Science 282 (1998) 706.

[12] G.J. Milburn, S.L. Braunstein, Phys. Rev. A 60 (1999) 937.[13] A. Jannussis, E. Skuras, Nuovo Cimento B 94 (1986) 29;

A. Jannussis, E. Skuras, Nuovo Cimento B 95 (1986) 63.[14] A. Jannussis, V. Bartzis, Nuovo Cimento B 100 (1987) 633[15] X. Ma, W. Rhodes, Nuovo Cimento B 104 (1989) 159.[16] E. Brändas, N. Elander (Eds.), Resonances, in: Lecture Note

in Physics, vol. 325, Springer-Verlag, Berlin, 1989.[17] A. Bohm, M. Gadella, Dirac Kets, Gamov Vectors and Gelfa

Triplets, in: Lecture Notes in Physics, vol. 348, SpringVerlag, Berlin, 1989.

[18] D. Chruscinski, J. Math. Phys. 45 (2004) 841.[19] K.W. Ford, D.L. Hill, M. Wakano, J.A. Wheeler, Ann. Phys.

(1959) 239.[20] W.A. Friedman, C.J. Goebel, Ann. Phys. 104 (1977) 145.[21] G. Barton, Ann. Phys. 166 (1986) 322.[22] N.L. Balazs, A. Voros, Ann. Phys. 199 (1990) 123.[23] M. Castagnino, R. Diener, L. Lara, G. Puccini, Int. J. Theo

Phys. 36 (1997) 2349.[24] T. Shimbori, T. Kobayashi, Nuovo Cimento B 115 (2000) 32[25] C.M. Caves, B.L. Schumaker, Phys. Rev. A 31 (1985) 3068[26] B.L. Schumaker, C.M. Caves, Phys. Rev. A 31 (1985) 3093[27] B.L. Schumaker, Phys. Rep. 135 (1986) 317.[28] D. Chruscinski, J. Math. Phys. 44 (2003) 3718.[29] C.G. Bollini, L.E. Oxman, Phys. Rev. A 47 (1993) 2339.[30] I.M. Gelfand, G.E. Shilov,Generalized Functions, vol.

Academic Press, New York, 1966.[31] R.P. Kanwal, Generalized Functions: Theory and Technique

in: Mathematics in Science and Engineering, vol. 177, Acemic Press, New York, 1983.

[32] K. Yosida, Functional Analysis, Springer, Berlin, 1978.[33] X. Ma, W. Rhodes, Phys. Rev. A 41 (1990) 4625.