arXiv:quant-ph/0403106v1 15 Mar 2004 orsodn ievcos hsosraini hngene then to is finally observation This eigenvectors. corresponding rsn ae eaegigt lrf hspolm npartic In problem. this clarify squee to the going of eigenvalues are proper we was that result paper stressed this present who of [12] interpretation al. the in However, et reported 11]. was corresp families [10, the these al. to of e One belong these not functions. Clearly, integrable do eigenvectors eigenvalues. corresponding real the discrete of families two htrsrcigt utbecaso tts(..coheren (e.g. states of scatt class of the suitable in action a known properties to well states restricting spectral resonant that its as to interpreted i related be does may closely spectrum are continuous purely which a values with unitary an lxpae oee,i snttewoesoy culy ti it Actually, story. whole the show th not to cover is easy it and However, is continuous It plane. purely plex is one. operator modulus operator with unitary squeeze numbers are complex operators of squeeze subset Clearly, operators. squeeze rsn ae eaayeteseta rpriso single- of properties serie spectral a the in analyze investigated we were states paper g these present the of by teleportation. reported properties quantum experiment ical with Rec the in connection exists. teleported in analog successfully classical interest no o of quantum which deal modern for the states in quantum role are prominent a play states Squeezed Introduction 1 hs ievle r lsl eae oteseta rpriso properties spectral the ge to real related discrete of closely family are a eigenvalues to rise These gives spectrum continuous purely h cteigter.I un u htteesae nieyc entirely states these that out turns resonan It as S theory. interpreted scattering be may the eigenvectors generalized sponding ( pcrlpoete ftesueeoperator squeeze the of properties Spectral z eso htasnl-oesueeoperator squeeze single-mode a that show We .Ti euti hngnrlzdto generalized then is result This ). N S ( md squeezing. -mode z a eetrl hrceie ntrso hs iceeei discrete these of terms in characterized entirely be may ) nttt fPyis ioasCpriu University Copernicus Nicolaus Physics, of Institute l rdiaza57 710Toru´n, Poland 87-100 Grudzi¸adzka 5/7, ul. ais Chru´sci´nskiDariusz Abstract N md squeezing. -mode 1 S ( z en nuiayoeao iha with operator unitary an being ) ttsd eogt hscas the class) this to belong do states t gnausaentpoe,ta is, that proper, not are igenvalues aie otomd qezn and squeezing two-mode to ralized h orsodn eigenvectors corresponding The . fppr 5 ,7 ,9.I the In 9]. 8, 7, 6, [5, papers of s o la i a rtczdb Ma by criticized was (it clear not opo uuaa[] Mathemat- [4]. Furusawa of roup eiso aesb ansi et Jannussis by papers of series a n ec hi pcr en a define spectra their hence and ayt oethat note to easy s eoeao onteit.I the In exist). not do operator ze mode nieui iceo h com- the on circle unit entire e nigHletsaeo square of space Hilbert onding nl,te aedonagreat a drown have they ently, rn hoy[3 4.W show We 14]. [13, theory ering de osse iceeeigen- discrete possesses ndeed lrw hwthat show we ular qezdsae flgtwere light of states Squeezed aatrz h cinof action the haracterize tc,seeg 1 ,3.They 3]. 2, [1, e.g. see ptics, ttswl nw in known well states t S f eaie eigenvalues. neralized httesetu fthe of spectrum the that ( S z ( and ) z n h corre- the and ) evle n the and genvalues N -mode S ( S z displays ) ( z being ) S N ( Z b ) 2 Single-mode squeezing

A single-mode squeeze operator is defined by

1 S(z) = exp za†2 z∗a2 , (2.1) 2h − i where z is a complex number and a (a†) is the photon annihilation (creation) operator which obey the standard commutation relation [a, a†] = 1. Clearly, S(z) may be represented as S(z) = exp(iH(z)), with 1 H(z)= za†2 z∗a2 . (2.2) 2i  −  Now, to investigate spectral properties of H(z) let us consider a unitarily equivalent operator R†(ϕ)H(z)R(ϕ), where R(ϕ) is a single-mode rotation [15, 16, 17]

R(ϕ) = exp(iϕ a†a) . (2.3)

One has R†(ϕ)H(z)R(ϕ) = H(ze−2iϕ) , (2.4) and hence, for ϕ = θ/2, where z = reiθ, it shows that H(z) is unitarily equivalent to H(r). Introducing two quadratures x and p via x + ip x ip a = , a† = − , (2.5) √2 √2 one finds the following formula for H(r): r H(r)= (xp + px) . (2.6) −2 Spectral properties of H(r) were recently investigated in [18] (see also [19]) in connection with quantum dissipation. Note, that the classical Hamilton equations implied by the Hamiltonian H = rxp: − x˙ = rx , p˙ = rp , (2.7) − describe the damping of x and pumping of p. This is a classical picture of the squeezing process. It turns out that H(r) has purely continuous spectrum covering the whole real line. Hence, it is clear that it does not have any proper eigenvalue. Using standard Schr¨odinger representation for x and p = id/dx, the corresponding eigen-problem H(r)ψ = Eψ may be − rewritten as follows d E 1 x ψ(x)= i + ψ(x) . (2.8) dx −  r 2

Note, that H(r) is invariant and hence each generalized eigenvalue E R is doubly ∈ degenerated. Therefore, two independent solutions of (2.8) are given by

E 1 −(iE/r+1/2) ψ±(x)= x , (2.9) √2πr ±

2 λ where x± are distributions defined as follows [20] (see also [21]):

xλ x 0 0 x 0 xλ := ≥ , xλ := ≥ , (2.10) +  0 x< 0 −  x λ x< 0 | | with λ C (basic properties of xλ are collected in [18]). These generalized eigenvectors ψE ∈ ± ± are complete

ψE(x)ψE (x′) dE = δ(x x′) , (2.11) Z ± ± − and δ-normalized

E1 E2 ψ (x)ψ (x) dx = δ(E1 E2) . (2.12) Z ± ± − Hence they give rise to the following spectral resolution of H(r):

H(r)= E ψE ψE dE , (2.13) Z | ± ih ±| X± and the corresponding spectral resolutions of squeeze operator S(r) immediately follows

S(r)= eiE ψE ψE dE . (2.14) | ± ih ± | X± Z Now, let us observe that FH = HF , where F denotes the Fourier transformation. Hence, E E − −E −E −E if H(r)ψ = Eψ , then H(r)F [ψ ] = EF [ψ ]. Therefore, the family F [ψ± ] defines another system of complete and δ-normalized generalized eigenvectors of H(r). Note that the action of S(r) is defined by

S(r)ψ(x)= e−r/2ψ(e−rx) , (2.15) and its Fourier transform F [S(r)ψ](p)= er/2F [ψ](erp) , (2.16) due to FS(r)= S( r)F , that is, if the fluctuations of p are reduced then the fluctuation of − x are amplified and vice versa.

3 Discrete real eigenvalues of S(r)

Surprisingly, apart from the continuous spectrum H(r) gives rise to the following families of complex discrete eigenvalues [18]

H(r)f ± = E f ± , (3.1) n ± n n where 1 E = ir n + , (3.2) n  2

3 and n n − ( 1) (n) + x fn (x)= − δ (x) , fn (x)= . (3.3) √n! √n! Interestingly they satisfy the following properties: ∞ + − fn (x) fm(x) dx = δnm , (3.4) Z−∞ and ∞ f +(x) f −(x′)= δ(x x′) . (3.5) n n − Xn=0 It implies that ± ±iEn ± S(r)fn = e fn , (3.6) which shows that S(r) displays two families of purely real generalized eigenvalues 1 s± = exp r n + . (3.7) n ±  2 + A family sn was already derived by Jannussis et al., see e.g. formula (2.3) in [11], but they − overlooked the second one sn . How to interpret these eigenvalues? It turns out that one ± E −E recover En and fn by studying a continuation of generalized eigenvectors ψ± and F [ψ± ] into the energy complex plane E C. Both ψE and F [ψ−E ] display singular behavior when E ∈ is complex: ψE has simple poles at E = E , whereas F [ψ−E] has simple poles at E =+E , ± − n ± n with En defined in (3.2). Moreover, their residues correspond, up to numerical factors, to the ± eigenvectors fn : Res(ψE (x); E ) f − , (3.8) ± − n ∼ n and Res(F [ψ−E (x)]; +E ) f + . (3.9) ± n ∼ n Such eigenvectors are well known in scattering theory as resonant states, see e.g. [13] and references therein. In the so called rigged Hilbert space to these states are also called Gamov vectors [14]. To see the connection with the scattering theory let us observe that under the following canonical transformation: rQ P rQ + P x = − , p = , (3.10) √2r √2r H(r) transforms into the unitarily equivalent operator [22] 1 H(r) H = (P 2 r2Q2) , (3.11) −→ io 2 − which represents the Hamiltonian of the called an inverted or reversed oscillator (or equiva- lently a potential barrier ‘ r2Q2/2’) and it was studied by several authors in various contexts − [23, 24, 25, 26, 27, 28]. An inverted oscillator Hio corresponds to the harmonic oscillator with a purely imaginary frequency ω = ir and hence the harmonic oscillator spectrum ‘ω(n+1/2)’ ± implies ‘ ir(n + 1/2)’ as generalized eigenvalues of H . ± io

4 4 A new representation of S(r)

± ± Interestingly, the action of S(r) may be entirely characterized in terms of fn and sn . Indeed, consider a space of smooth functions ψ = ψ(x) with compact supports, i.e. ψ(x) =0 for D x > a for some positive a (depending upon chosen ψ), see e.g. [29]. Clearly, defines a | | D subspace of square integrable functions L2(R). Moreover, let = F [ ], that is, ψ if Z D ∈ Z ψ = F [φ] for some φ . It turns out [29] that and are isomorphic and = . ∈ D D Z D∩Z ∅ Now, any function φ from may be expanded into Taylor series and hence Z ∞ ∞ φ(n)(0) φ(x)= xn = f +(x) f − φ . (4.1) n! n h n | i nX=0 nX=0 On the other hand, for any φ , its Fourier transform F [φ] , and ∈ D ∈Z ∞ 1 1 F [φ](n)(0) φ(x) = eikxF [φ](k)dk = eikx kn dk √ Z √ Z n! 2π 2π nX=0 ∞ ∞ = F [f +](x) f − F [φ] = f −(x) f + φ . (4.2) n h n | i n h n | i nX=0 nX=0 Hence, we have two decompositions of the identity operator

∞ 1l= f + f − on , (4.3) | n ih n | Z nX=0 and ∞ 1l= f − f + on . (4.4) | n ih n | D nX=0 It implies the following representations of the squeeze operator S(r):

∞ S(r)= s− f + f − on , (4.5) n | n ih n | Z Xn=0 and ∞ S(r)= s+ f − f + on . (4.6) n | n ih n | D nX=0 It should be stressed that the above formulae for S(r) are not spectral decompositions and they valid only on and , respectively (its spectral decomposition is given in formula Z D (2.14)). Note, that S(r) maps into and using (4.5) one has Z D ∞ S†(r)= s− f − f + = S( r) on . (4.7) n | n ih n | − D nX=0

5 Conversely, S(r) represented by (4.6) maps into and S†(r) = S( r) on . It shows D Z − Z that squeezing of x (p) corresponds to amplifying of p (x). Clearly, a general quantum state ψ L2(R) belongs neither to nor to . An example of quantum states belonging to ∈ D Z Z is a family of Glauber coherent states α . Consider e.g. a squeezed vacuum S(r)ψ0, where −1/4 −x2/2 | i ψ0(x)= π e . One has

∞ n 1 ( 1) + ψ0(x)= − f (x) , (4.8) π1/4 2n nX=0 (2n)! p and −r/2 ∞ −2r n −r/2 ∞ n e ( e ) + e ( 1) + −r S(r)ψ0(x)= − f (x)= − f (e x) . (4.9) π1/4 2n π1/4 2n Xn=0 (2n)! nX=0 (2n)! p p The similar formulae hold for S(r) α . | i 5 Two-mode squeezing

Consider now a two-mode squeeze operator [15, 16, 17]

† † ∗ S2(z) = exp za1a2 z a1a2 , (5.1)  −  † where ak and ak are creation and annihilation operators for two modes k = 1, 2. Introducing 2-dimensional vectors T † T † † a = (a1, a2) , (a ) = (a1, a2) , (5.2) one finds † T † ∗ T S2(z) = exp z(a ) σ1a z a σ1a , (5.3)  −  where σ1 stands for the corresponding Pauli matrix. Using well-known relation

iπ/4 σ2 −iπ/4 σ2 e σ1 e = σ3 , (5.4) one obtains

iπ/4 σ2 −iπ/4 σ2 † T † ∗ T e S2(z) e = exp z(a ) σ3a z a σ3a  −  = S(1)(z)S(2)( z) , (5.5) − where S(k) denotes a single-mode (k) squeeze operator. Now, since S(k)(z) is unitarily equiv- alent to S(k)(r) a two-mode squeeze operator S (z) is unitarily equivalent to S(1)(r)S(2)( r). 2 − Hence, the spectral properties of S (z) easily follows. In particular defining the space 2 D of smooth functions ψ = ψ(x ,x ) with compact supports and = F [ ] one obtains the 1 2 Z D following representations:

∞ S(1)(r)S(2)( r)= s− f + f − on , (5.6) − nm | nm ih nm| Z nmX=0

6 and ∞ S(1)(r)S(2)( r)= s+ f − f + on , (5.7) − nm | nm ih nm| D nmX=0 where ± ± ± fnm(x1,x2)= fn (x1)fm(x2) , (5.8) and ± ±r(n−m) snm = e . (5.9) 2(m−n) Jannussis et al. [11] claimed that the eigenvalues of S2(z) are given by e , see e.g. − formula (5.7) in [11]. Their result has the similar form as snm but of course it is incorrect. Note that eigenvalues of [11] do not depend upon the squeezing parameter z as was already observed in [12].

6 N-mode squeezing

Following [30] one defines an N-mode squeeze operator

1 1 S (Z) = exp (a†)T Z a† a† Z† a (6.1) N 2 − 2  b b b where Z is an N N symmetric (complex) matrix and × b T a = (a1, a2, . . . , aN ) .

Defining an N-mode rotation operator

† T RN (Φ) = exp i(a ) Φ a , (6.2)   b b with Φ being an N N hermitian matrix, one shows [17, 30] × b † −iΦ −iΦT RN (Φ) SN (Z) RN (Φ) = SN e Z e . (6.3)  b b  b b b b Now, by a suitable choice of Φ one obtains

b −iΦ −iΦT e Z e = ZD , (6.4) b b b b where ZD is a diagonal matrix, i.e. (ZD)kl = zkδkl. Hence, an N-mode squeeze operator S (Z) is unitarily equivalent to N b b b † (1) (2) (N) RN (Φ) SN (Z) RN (Φ) = S (z1) S (z2) ... S (zN ) , (6.5) and therefore its propertiesb areb entirelyb governed by the properties of the single-mode squeeze operator S(z). In particular SN (Z) gives rise to a discrete family of generalized eigenvalues b

7 (k) being combinations of generalized eigenvalues of S (zk). Defining the corresponding sub- spaces and in the Hilbert space L2(RN ) one easily finds D Z ∞ R† (Φ) S (Z) R (Φ) = s− f + f − on , (6.6) N N N n1...nN | n1...nN ih n1...nN | Z n1,...,nXN =0 b b b and ∞ R† (Φ) S (Z) R (Φ) = s+ f − f + on , (6.7) N N N n1...nN | n1...nN ih n1...nN | D n1,...,nXN =0 b b b where ± ± ± fn1...nN (x1,...,xN )= fn1 (x1) ...fnN (xN ) , (6.8) and 1 1 s± = exp r n + + . . . + r n + , (6.9) n1...nN ±  1  1 2 N  N 2 with r = z . k | k| Acknowledgments

This work was partially supported by the Polish Ministry of Scientific Research and Infor- mation Technology under the grant No PBZ-MIN-008/P03/2003.

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