Quantum noise of modulation and squeezed states

Quantum noise of modulation and squeezed states

Volume 119, number 2 PHYSICS LETFERS A 1 December 1986 QUANTUM NOISE OF MODULATION AND SQUEEZED STATES Y. BEN-ARYEH Department of Physics, Technion...

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Volume 119, number 2

PHYSICS LETFERS A

1 December 1986

QUANTUM NOISE OF MODULATION AND SQUEEZED STATES Y. BEN-ARYEH Department of Physics, Technion — IsraelInstitute of Technology, Haifa 32000, Israel Received 28 August 1986; accepted for publication 9 October 1986

The reduction of the amount of quantum noise in modulation experiments by the use oftwo-mode squeezed states is discussed. It is shown that by the use of squeezed radiation instead of coherent radiation, the amount of quantum noise at the modulated frequency can be reduced by many orders of magnitude.

In the theory of squeezed states one has to distinguish between squeezed states in one mode of radiation [1], and those which are related to two modes of radiation [2]. Squeezed states have been defined as minimum uncertainty states in which the minimum uncertainty in one quadrature is small relative to the other quadrature [31. However, under the hamiltonian of free space a squeezed state in one mode of radiation does not develop as a minimum uncertainty state and the minimum uncertainty is oscillating between the two quadratures [1,4]. Therefore, the reduction of noise in one-mode squeezed states is related to the number operator and to the measurement of subpoisson photon statistics [3,5—71. For a strong coherent radiation the poisson distribution is already very narrow and although the possibility to make it narrower is oftheoretical interest, we do not expect any strong effects which will result from the squeezing process. The purpose of this letter is to present a new theoretical explanation to the modulation experiments with two-mode squeezed radiation [8—10].In these experiments, one uses two-mode squeezed states of radiation with frequencies Q±~ and filters the total signal by a spectrum analyzer to pick out the contribution from a single modulation frequency (e’i~zQ). The two-mode squeezed states are obtained by nonlinear optical processes [8—10] in which the quanturn noise is coupled between the two modes of radiation [2]. In the interference between the twomode squeezed radiation and a local oscillator at fre0375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

quency Q, one observes strong fluctuations in the amount of quantum noise as a function of the phase of a local oscillator [8—10]. I explain here these experiments by relating them to the properties oftwomode coherent and squeezed radiation. The two-mode coherent states can be created from the vacuum state by applying separate “displacement operators” to the vacuum state [11]: a+, a_ >=D(a+, a±)D(a~, a_)lO>,

(1)

where D(a~,a~)D(a_,a_) =exp(a~a~—a~+a_at —a~a_).

(2)

These states are eigenstates of the annihilation operators a÷and a_ for the two modes at frequencies ~and Q— e with complex eigenvalues a + and a_, respectively. The two-mode squeezed states are given by

I a +, a > r,ø = S( r, 0)1 a +, a > ~ —



(3)

where the unitary two-mode squeeze operator is given by S(r, Ø)=exp[r(a±a_ e 2~0—a~a~e210)]. (4) The realnumber r is called the “squeeze factor” while 0 is a phase angle. Such states are generatedfrom twomode coherent states by two-photon processes [8—10]. The most important property of the twomode squeeze operator which is used in this letter is given by [2] 51

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PHYSICS LETTERS A

St(r, Ø)a+ S(r, 0) =a±cosh r—a~,e2’0 sinh r.

(5)

Let us consider the heterodyning interference

I December 1986

classical field for the interfering radiation which oscillates at frequency Q. Following these considerations, we calculate the quantum fluctuations at frequency e by relating them to the operators E~and E

2.

experiments [8—10]in which a plane coherent wave at frequency Q is superimposed on a plane two-mode squeezed wave with frequencies Q± We decompose the electromagnetic field operator E into positive and negative parts: E(x,t)=E~(x,t)+E~~(x,t). (6)

Therefore, we represent the total intensity of the radiation operator (ignoring proportionality constant) as

2 ={E

~.

For the two modes at frequencies Q ±e we have: ~



(w —~)~2a e _i(Q_>(t_x/c)],

E~~(x,t)= [ Et~~(x, t)]~.

1 cos[Q(t—x/c)] +E2 sin[Q(t—x/c)]

2. (10) +Bsin[Q(t—x/c)+ô]} By averaging over the rapid oscillations we get

<>

~(x, t) =2 ~“2[(Q+e) “2a÷e —j(Q+~)(t—-x/c) +

I=E

=

(7)

<>

=

1/2

<>

For treating the modulation quantum noise in the heterodyning experiments it is convenient to decompose the electric field at the two modes Q ±e into its quadrature phases:

= <> =0, (11) where << >> indicates here an ensemble averaging over times which are large relative to the time period ofoscillation. We get therefore:

E(x, t) =E~(x,I) cos[Q(t—x/c)]

I_—r~E~+~E~ +~B2+EIB sin o+E 2B cos ~

(12)

+E2(x, t) sin[Q(t—x/c)], E~(x,t) =E~+ ~(x, t)e”~’~”~

The operators Em (m= 1 or 2) can be expressed as [2]

+E~~(x, t)e_(~>,

Em

E2(x, t)=—iE~~(x, t)el~tK~c~)

+iE~~(x, t)etQ(t~~. (8) E 1 and E2 are operators which oscillate at fre-

=

[ame

—it—K/C)

+a~~e’’~’~],

2(2÷a++..La~)

a~=2~ a2 = 2



1/2

( — iA + a ± + iA_ a

quency ~ and are the natural operators for treating modulation effects. We superimpose on the two-mode squeezed wave E a classical carrier wave at frequency Q given by

A ±=

E~1=B sin[Q(t—x/c) +ô],

I(e) =E1B sin o+E2B cos .5.

(9)

where B is real, and ô is a phase angle of the classical oscillator which can be changed in the experiments, The assumption of a classical field for the carrier wave is justified for the analysis of interference experiments in which the intensity of radiation and its fluctuations are measured at the modulated frequency e. Since the time scale in these measurements is very large relative to Q’, the quantum fluctuations at frequency Q are averaged to zero by the measurement process. We may, therefore, assume a 52

(13)

The component of I which oscillates at the modulated frequency e is given by (14)

We ignore here all the terms of I [eq. (12)] which do not oscillate, or oscillate at a frequency different from e, in agreement with the selection of frequency ~by the spectrum analyzer. The quantum fluctuations at frequency e are described by: M(e)=

[

— <‘(i) >21 1/2

(15)

We get, therefore, the result that the measured quantum noise at the modulated frequency ~ is related to the uncertainties in the two quadratures of the elec-

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PHYSICS LETTERS A

<(LS.E~)2>~Q(cosh r—sinh r)2

tromagnetic field: AJ(~)=B[<(AE 2> sin2ô+<(i~..E 2>cos2ô 1) 2) + sjfl By straightforward calculations we get according to the above equations <(AE1 )2> = r,Ø r,Ø —







2

=Q(cosh2 r+ sinh2 r) /

2

2

(17)

2> = ~.ø =Q(cosh2r+sinh2r)

<(~E —

sinh rcosh rcos 20,

(18)

‘i.VEI E21/ \

<(~.E 2>i~Q(coshr+sinh r)2 2) (e2/Q) sinh r cosh r, <~(E 1E2)> =0. —

(21)

The noise <(L~.E1)2> in one of the quadratures in the second quadrature becomes very large (the2> opposite becomes very small while the noise <(AE2) situation occurs for cos 20 = 1). We find according to eq. (16) that by changing in the experiments the phase angle ö the amount of noise M( ) oscillates between the two values of noise corresponding to these two quadratures. By changing the squeeze parameter r the mini2> isgivenby mum noiseobtainedfor <(iIEi) <(AE 2>~e (fore2r=2Q/e). 1) The quantum noise of modulation obtained by squeezed states can be reduced by a maximal factor Q/e relative to the quantum of modulation produced by two-mode coherentnoise radiation.

‘.

=~
+(~2/Q)sinhrcoshr,



,

— 2~Q — ~ sinh r cosh r cos 20,

+2~~/Q2 ~2

1 December 1986

1E

1E2-i-E2E1 Ia+, a_ >r,Ø

r,Ø = —2,.JQ2 _~2 sinh rcosh rsin 20.

(19)

The coherent two-mode state of radiation may be considered as a special case in our analysis for which cosh r= 1, sinh r= 0. For this case we get 2> ~ ((i~.E1)2 > = < (i.S.E2) = 0.

(20)

For the two-mode coherent state of radiation the quantum noise atbetween the modulated frequency The e is equally distributed the two quadratures. quantum noise Q (h = 1) is very large relative to the quantum noise d2 which is the quantum noise of a signal sent directly at frequency In orderto illustrate the properties of the two-mode squeezed states in the heterodyne experiments we ~.

discuss the two specific cases: (a) cos 20= 1, (b) sin 20 = 0. Since in these_experiments ~~ £~,we use the approximation ,,/Q2 ~ Q_ ~2/2Q. For the first case (cos 20= 1) we get

Forthecase sin 20=1 weget 2> = <(AE 2> =Q(cosh2r+sinh2r), <(z~E1) 2) =2(Q—~ Although in the present case the amount of noise at the two quadratures is equal, the fluctuations in the total amount of noise at frequency ~are produced by the coupling term . amount of noisetowhich are similar to those obtained in the first case. The minimum amount ofnoise is obtained when sin 2ô =1 and then we get 2 A1(e) =B[Q(cosh r—sinh1/2r) 2 +(e /Q)sinhrcoshr] ,

(23)

which is equivalent to the minimum amount ofnoise obtained for in the first case. The treatment of the general case (for any phase angle 0) leads to somewhat more complicated expressions which will not be of interest as they give the same basic physical results. According to the theory presented in this letter, the use of two-mode coherent radiation with frequencies 53

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Q ±~ for producing a signal at the modulated frequency (~<
54

1 December 1986

References [1] Y. Ben-Aryeh and A. Mann, Phys. Rev. A 32 (1985) 552; Phys. Rev. Lett. 54 (1985) 1020. [2] CM. Caves and B.L. Schumaker, Phys. Rev. A 31(1985) 3068; B.L. Schumaker, Phys. Rep. 135 (1986) 317. [3] D.F. Walls, Nature 306 (1983) 141. [4] R.A. Fisher, N.M. Nieto and V.D. Sandberg, Phys. Rev. D 29 (1984) 1107. [5] R. Loudon, Opt. Commun. 49 (1984) 67. [6]Y.Ben-AryehandA.Mann,Phys.Lett.A112(1985)371. [71L. Mandel, Phys. Rev. Lett. 49 (1982) 136. [8] RE. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz and J.F. Valley, Phys. Rev. Lett. 55 (1985) 2409. [9] M.W. Maeda, P. Kumar and J.H. Shapiro, Observation of squeezed noise produced by four wave mixing in sodium vapor, post deadline paper at Intern. Quantum Electronic Conf., San Francisco (1986). [101 H.J. Kimble and J.L. Hall, Generationof squeezed states of light by intra-cavity harmonic conversion, Invited Lecture at the Intern. Quantum Electronic Conf., San Francisco (1986). [11] R.J. Glauber, Phys. Rev. 131 (1963) 2766.