Sampling quantum states via heterodyne measurements with non-classical probe states

Sampling quantum states via heterodyne measurements with non-classical probe states

1 January 1999 Optics Communications 159 Ž1999. 158–168 Full length article Sampling quantum states via heterodyne measurements with non-classical ...

1MB Sizes 0 Downloads 30 Views

1 January 1999

Optics Communications 159 Ž1999. 158–168

Full length article

Sampling quantum states via heterodyne measurements with non-classical probe states Dietmar G. Fischer 1, Matthias Freyberger Abteilung fur ¨ Quantenphysik, UniÕersitat ¨ Ulm, D-89069 Ulm, Germany Received 26 June 1998; revised 13 August 1998; accepted 11 September 1998

Abstract We describe an intrinsically non-classical method for the reconstruction of the quantum state defining the signal mode of a heterodyne set-up. This is achieved by mixing the signal mode with a probe mode prepared in a Schrodinger-cat state. The ¨ emerging quantum interferences in the output photocurrent allow us to map out the state of the signal mode. We emphasize that the interference term actually serves as a complete phase representation of a quantum state leading to a simple reconstruction algorithm. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction Quantum objects are the most fragile and at the same time most fundamental components of the physical nature that we are aware of. Any single observation event changes the state of a quantum object w1x and therefore its future behavior. This also implies that measurements performed on a single quantum object will not uncover its state completely. But having an ensemble of identically prepared quantum systems it is actually possible to find out what the underlying quantum state is. This works since we can now measure the statistics of a complete set of observables which allows us to reconstruct the state. The point is then to find the proper set of observables that is complete in the sense that its statistics encodes the full quantum-state information. Which and how many observables must be measured is then the theoretically challenging question and it was Wolfgang Pauli who first addressed it in his famous handbook article as early as 1933 w2x. The past few years have seen an enormous growth of interest in this considerably old problem of quantum state measurements. For an overview of the wide-ranging theoretical achievements in this field we refer the reader to the reviews in Refs. w3,4x and to the special issue on quantum state preparation and measurement w5x. It is however worthwhile to have a closer look at the intriguing experimental results in this field since they certainly triggered the aforementioned explosion of interest. The tomographic principle w6,7x has been realized in optical experiments w8,9x that revealed the vacuum state as well as squeezed states of a quantized light mode. These tomographic experiments are based on the measurement of a complete set of incompatible observables which in the optical regime can be realized via sophisticated four-port homodyne techniques. In contrast to that one can also use eight-port homodyne methods to probe the joint statistics of a set of compatible observables. Early experiments along these lines have been reported in Ref. w10x. The experimental art of quantum state measurements was not only developed for quantized light fields but also for quantum states of matter. Quantum state tomography was again the guiding principle that lead to the first experimental reconstruction of a quantum state describing the vibrational motion of a diatomic molecule w11x and it was only recently that

1

E-mail: [email protected]

0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 4 9 2 - 1

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

159

the same principle was applied to a beam of atoms impinging on a double slit w12x. For a trapped ion an interesting alternative method to tomography was used showing for the first time experimentally measured negative parts in the Wigner representation of a non-classical quantum state w13x. For a further discussion of the basic principles of all these approaches we refer the reader to Refs. w14,15x. In the present paper we show a reconstruction method that is particularly suited to probe quantized light of the signal mode of a heterodyne set-up. However, heterodyne detection can be shown w16x to be equivalent to eight-port homodyning. Thus the reconstruction method that we are going to describe can also be realized in an eight-port homodyne set-up. Our main emphasis lies on the intrinsically non-classical character of the proposed scheme. We achieve this by mixing the signal mode with a probe mode that has been prepared in a suitably designed non-classical quantum state, namely a Schrodinger-cat ¨ state. We shall show that this leads to quantum interference which is measurable and in addition carries the complete information on the quantum state of the signal mode. In addition, the quantum state can be mapped out easily in Fock representation by simply Fourier analyzing the quantum-interference terms. The paper is organized as follows. In Section 2 we shortly review the well-known basic properties of the heterodyne set-up using the language of positive operator measures ŽPOM. w17,18x. In Section 3 this POM description turns out to be especially useful for an elegant formulation of the problem when we shine non-classical light into the probe mode. The specific method to map out the quantum state from the measured statistics is described in Section 4 and we simulate examples in Section 5. The final Section 6 contains our conclusions.

2. The POM for the heterodyne measurement Heterodyne detection of electromagnetic signals is a well-established method which is applicable over a broad range of frequencies. Here we will only state the main results of the quantum mechanical analysis of heterodyne detection given in Ref. w19x. The basic element of a heterodyne set-up shown in Fig. 1 is a beam splitter ŽBS. with transmittivity u . From the left side of the beam splitter the two important light modes enter: the signal mode s at a frequency v 0 q v X and a probe mode p at a frequency v 0 y v X. We assume that the signal mode is in a state described by the density operator rˆs and that the state of the probe mode is given by a pure state < c : p . Signal mode and probe mode are mixed with a local oscillator mode of frequency v 0 which has been prepared in the coherent state < z :. The outgoing time-dependent light intensity is Fourier analyzed at the intermediate frequency v X and the corresponding Fourier component can be described by an operator Iˆout Ž v X .. In the limit of a strong local oscillator < z < ™ ` and a transmittivity u ™ 1 we can measure w19x the scaled operator lim u ™1, < z <™` g sconst

Iˆout Ž v X .

g

ˆ s aˆs q aˆ†p ' A,

Ž1.

if g ' < z < u Ž 1 y u . is kept constant. The operators aˆs and aˆ†p are annihilation and creation operators of signal and probe mode respectively.

'

Fig. 1. Schematic set-up of the heterodyne measurement. From the left light of the signal mode s and probe mode p enters the measurement X X apparatus. The frequencies of signal and probe mode are centered at v 0 q v and v 0 y v respectively. At the beam splitter ŽBS. these light modes are mixed with a local oscillator of frequency v 0 prepared in the coherent state < z :. The detector effectively measures the X X Fourier component Iˆout Ž v . of the intensity of the outgoing light centered at the intermediate frequency v .

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

160

Measuring the operator Aˆ actually corresponds to the measurement of a pair of commuting Hermitian operators Aˆr s 12 Ž Aˆq Aˆ† . and Aˆi s 21i Ž Aˆy Aˆ† .. Hence we can construct a joint probability density p˜ Ž a r ,a i . for the measurement results Ž a r ,a i . of these operators. However, notation is considerably simpler if we use the probability density pŽ a,a ) . of the operators Aˆ and Aˆ† for our calculations. This probability density can be written as p Ž a,a ) . s

d 2l

Hp

2

ˆ Trs ,p rˆs m < c : p p² c < m e l A



y l ) Aˆ

el

)

ay l a )

.

Ž2.

It has been shown in Ref. w20x that this probability density can also be expressed in the form p Ž a,a ) . s

1

p

² c˜ < Dˆs† Ž a . rˆs Dˆs Ž a . < c˜ : s ,

Ž3.

s



with the definition of the displacement operator DˆsŽ a. s e aaˆsya

)

aˆs

and the state

`

< c˜ : s s

n Ý Žy1. p² n < c :)p < n: s .

Ž4.

ns 0

Note that < c˜ : s is defined by the Fock coefficients p² n < c : p of the probe light. The comparison of this result with the definition w17,18x p Ž a,a ) . d 2 a ' Trs rˆs d Pˆ s Ž a,a ) .

Ž5.

of a POM d Pˆ sŽ a,a ) . yields d Pˆ s Ž a,a ) . s

d2a

p

Dˆs Ž a . < c˜ : s s² c˜ < Dˆs† Ž a . .

Ž6.

If the probe mode is in the vacuum state < c : p s <0: p , we find the POM w19x d Pˆ s Ž a,a ) . s

d2a

p

< a: s s² a < ,

Ž7.

with the coherent state `

< a: s Dˆ Ž a . <0: s ey< a<

2

r2

an

Ý ' ns 0 n!

< n: .

Ž8.

This means that with the vacuum as probe state the joint probability density p Ž a,a ) . s Trs rˆs

1

p

< a: s s² a < s

1

p

² a < rˆs < a: s ' Qs Ž a .

s

Ž9.

reduces to the Q function QsŽ a. of the signal mode. In principle the Q function is a complete representation of the state of the system and we could reconstruct the density operator from it. This reconstruction scheme, however, is quite complicated. In the following we will generalize the POM, Eq. Ž6. and show that there exists a simpler reconstruction scheme, if we use non-classical cat-like probe states. We emphasize that by reconstruction we here mean the mapping of experimental data onto the Fock representation of the signal density operator.

3. The POM for cat-like probe states In the last section we have introduced the POM language for a measurement where just the quantum vacuum fluctuations enter the probe mode. It turns out that in this case the POM determining the measured joint probability density pŽ a,a ) . is diagonal in the coherent state basis. Due to the over-completeness of coherent states it turns out that such a diagonal representation Žthe Q function. of the state of the signal mode indeed contains the complete quantum state information 2.

2 Besides the Q function other complete representations of states in terms of coherent states exist as well. In this context special interest has been paid to one-dimensional coherent-state representations w21–23x.

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

161

However, unraveling this information in detail can be a complicated task. The question therefore arises whether we can modify the heterodyne measurement in a way to simplify the reconstruction of the signal mode’s quantum state. An obvious modification of the heterodyne set-up would be the application of suited probe light. In this section we shall discuss one possible candidate for a more general probe state, namely a superposition of two coherent states often called a Schrodinger-cat state. The corresponding POM for the signal mode will then consist of ¨ non-diagonal terms in the coherent state basis. Hence we shall see interference terms in the joint probability density governed by such a generalized POM of the heterodyne set-up. In the course of the paper it will become clear that these interference terms lie at the very heart of the reconstruction scheme. We now consider the special probe state < c : p s 'N < y a ) : p q < y b ) : p ,

Ž 10. 1 2

.y1

namely a coherent superposition of two coherent states < y a p and < y b p where N s Ž1 q Re p² a < b : p is the normalization constant. As this is a non-classical state made up of two quasi-classical coherent states, it is often called a Schrodinger cat state w24–26x. Using the minus signs and complex conjugation in the two coherent states of Eq. Ž10. just ¨ simplifies the notation in the following steps of the calculation. When we insert Eq. Ž10. in Eqs. Ž4. and Ž6. we find the POM ):

d Pˆ s Ž a,a ) . s

Nd 2 a

p

):

Dˆs Ž a . Ž < a : s q < b : s . Ž s² a < q s² b < . Dˆs† Ž a . ,

Ž 11.

which can be evaluated with the help of the relation )

Ds Ž a . < a : s s e iIm Ž a a . < a q a : s .

Ž 12.

The total POM for our cat-like probe state is therefore given by d Pˆ s Ž a,a ) . s

Nd 2 a

p

< a q a : s s² a q a < q < a q b : s s² a q b <

qe iIm w a Ž a

)

yb ) . x <

a q a : s s² a q b < q e i I m w a Ž b

)

ya ) . x <

a q b : s s² a q a < .

Ž 13.

It consists of two terms that are diagonal in the coherent state basis. The other two non-diagonal terms will give rise to interference terms in the probability density pŽ a,a ) ., Eq. Ž5., which is now based on this new POM of the heterodyne set-up with non-classical probe states. We shall show in the next section that the corresponding probability density allows us to extract the complete density operator rˆs of the signal mode in a very simple way.

4. Reconstruction of a density operator For any reconstruction method one has to choose a specific representation of the corresponding density operator. This is not a straightforward task. The mapping of the experimentally measurable quantities, in our case the probabilities pŽ a,a ) ., depends on this representation. The proper choice of a representation can therefore considerably simplify the mathematics of a mapping procedure. In our case the probabilities p Ž a,a ) . d 2 a s Tr rˆs d Pˆ s Ž a,a ) .

Ž 14 .

are most naturally mapped on the Fock-basis elements rn m ' s² n < rˆs < m: s of the density operator

rˆs s

Ý rn m < n: s s² m < .

Ž 15.

n,m

Using the Fock representation of rˆs and the POM, Eq. Ž13., we now concentrate on the probability density p Ž a s 0,a ) s 0 < a , b . s

N

p

² a < rˆs < a : s q s² b < rˆs < b : s q s² b < rˆs < a : s q s² a < rˆs < b : s

s

Ž 16.

at the point a s a ) s 0, which still depends on the setting of the coherent reference amplitudes a and b , now explicitly included in the notation.

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

162

To bring out the structure of Eq. Ž16. more clearly we rewrite it with the help of the Q function, Eq. Ž9., and by defining the interference term I Ž a , b . ' Re Ž s² a < rˆs < b : s .

Ž 17.

whereby we get p Ž 0,0 < a , b . s N Qs Ž a . q Qs Ž b . q

2

p

IŽ a ,b . .

Ž 18.

It is the interference term, Eq. Ž17., which lies at the very heart of our reconstruction method. In order to see this, we choose X a s re i f and b s re i f with a fixed but arbitrary radius r. That is, we assume that the amplitudes a and b can be varied independently on a fixed circle in the phase space of coherent states. Hence the interference term becomes a function of f and f X which is 2p-periodic in both variables. X The corresponding Fourier decomposition of I Ž re i f ,re i f . indeed allows us to reconstruct the elements rnm of the density operator via

rnm s Ž 2 y dn0 dm 0 . e r

s Ž 2 y dn0 dm 0 . e

2

'n!m! r nqm

r2

'n!m! r nqm

p

df

Hyp 2p p

df

e i nf

i nf

d fX

p

yi m f

X

I Ž re i f ,re i f .

yi m f

X

Re Ž s² re i f < rˆs < re i f : s . .

Hy p 2p e d fX

p

Hyp 2p e Hy p 2p e

X

X

Ž 19.

X

X

Hence I Ž re i f ,re i f . contains the complete information on the state rˆs , or, in other words, the real quantity I Ž re i f ,re i f . is a complete phaseX representation of the density operator rˆs w27x. Note that the mapping of the experimentally accessible data pŽ0,0 < re i f ,re i f . on the elements of the density operator is now really a straightforward task, since it basically consists in a simple Fourier analysis. Due to the fact that the measurable probability, Eq. Ž18., does not only depend on the interference term but also on two Q functions, the full reconstruction method is just slightly more sophisticated than Eq. Ž19.. We propose the following measurement strategy: X The probabilities pŽ0,0 < a , b . are measured for many phases f and f X of the coherent states a s re i f and b s re i f . We assume that the radius r is kept constant during this sampling process and that the phases are varied equidistantly in the interval wyp ,p x. The resulting probabilities are given by

p N

X

p Ž 0,0 < re i f ,re i f . s eyr

2

r mqn

X

X

X

Ý rnm 'm!n! we i f Ž myn . q e i f Ž myn . q e i m fyi n f q eyi n fqi m f x .

Ž 20.

n,m

In the ideal case we have phases lying dense on the radius r and hence we can calculate the Fourier integrals with respect to f and f X resulting in the Fourier components df

p

Mk l '

d fX

p

X

Hyp 2p Hy p 2p e `

s eyr

2

rnm

Ý n , ms0

r

iŽ k f yl f .

p N

X

p Ž 0,0 < re i f ,re i f .

mqn

'm!n!

w d k , nym d l ,0 q d k ,0 d l , myn q d k ,y m d l ,y n q d k , n d l , m x ,

Ž 21.

with k G 0 and l G 0. Eq. Ž21. really is the central reconstruction equation from which the elements rnm of the density matrix can be extracted easily. Due to the Kronecker d several cases have to be distinguished: Ži. k ) 0 and l ) 0: In this case only the last term of Eq. Ž21. is different from zero. Thus the density matrix elements are directly proportional to the corresponding elements of the Fourier integral Mk l s eyr Žii.

2

r kql

'k!l!

rkl .

Ž 22.

k s l s 0: All four terms contribute and we get ` 2

M00 s eyr 2

Ý ns 0

rn n

r 2n n!

` 2

q 2 r 00 s eyr 4 r 00 q 2

Ý ns1

rn n

r2n n!

.

Ž 23.

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

163

As the matrix elements r 11, r 22 , ... are already known from Ži., r 00 can be calculated. Žiii. k ) 0, l s 0: Only the first and the last term of Eq. Ž21. are non-zero so that `

Mk 0 s eyr

2

r 2 nqk

Ý rnqk , n 'n! '

ns 0

Ž nqk.!

q rk 0

rk

'k!

2

s eyr 2 r k 0

rk

`

q

'k!

r 2 nqk

Ý rnqk , n 'n! '

ns1

Ž nqk.!

.

Ž 24. Again all the matrix elements except r k 0 have already been calculated and therefore r k 0 can be determined. ) Živ. k s 0, l ) 0: Because of the relation rnm s rmn the matrix elements r 0 l are known as soon as the matrix elements r k 0 have been reconstructed. Thus we have indeed found a method for completely reconstructing density matrices by simply Fourier analyzing the measurable probabilities.

5. Numerical examples To illustrate our method we now simulate the reconstruction of two example states. In particular, we shall discuss the two most important factors influencing the quality of reconstruction 3. First, we need in principle an infinite number of phases f and f X of the probe states to evaluate the Fourier integrals in Eq. Ž21.. In experiments, however, only a finite number M of phases can be used. Hence the following numerical examples will focus on the number of measurements 4 influencing the quality of the reconstruction. Second, we shall show how the radius r of the measurement circle comes into play. Of course, looking at Eqs. Ž21. – Ž24., the radius r can be chosen at will. However, as soon as we have the aforementioned discretization of the phase values, there will be an influence of the radius r. As our first example we consider the reconstruction of the pure state < c Ž1. : s s 'N w < g 1 : s q < g 2 : s x ,

Ž 25.

which consists of two coherent states < g i : s with g 1 s 3e 0.2 i , g 2 s ey0.5i and N s 12 Ž1 q Re s²g 1 < g 2 : s .y1. The corresponding density-matrix elements

rnŽ1.m s s² n < c Ž1. : s s² c Ž1. < m: s

Ž 26.

are shown in Fig. 2Ža.. Furthermore, in contrast to this coherent superposition, we also analyze an incoherent superposition of < g 1 : s and < g 2 : s whose elements of the density matrix are given by

rnŽ2.m s 12 Ž s² n < g 1 : s s²g 1 < m: s q s² n < g 2 : s s²g 2 < m: s . .

Ž 27 .

We illustrate them in Fig. 3Ža.. It is obvious from a comparison of Figs. 2Ža. and 3Ža. that the coherent superposition has additional oscillations in the Ž1. showing the coherence properties of the state rˆ Ž1.. This difference between the coherent and non-diagonal parts of rnm incoherent superposition can be seen as well in I Ž a , b ., Eq. Ž17., which, as discussed in the preceeding section, is a complete representation of rˆs . The real quantities I Ž1. representing the coherent superposition rˆ Ž1. and I Ž2. representing the incoherent counterpart rˆ Ž2. are plotted in Figs. 4Ža. and 4Žb., respectively. The coherence properties of rˆ Ž1. are clearly visualized by the corresponding oscillatory behavior of I Ž1. close to the origin Ž f , f X . s Ž0,0.. We now study the influence of the phase discretization on the reconstructed density-matrix elements rnm. For our simulation we first of all calculate the probability pŽ0,0 < a , b ., Eq. Ž18., for M = M probe states, that is, for M = M

3

For this discussion we concentrate on the most fundamental factors which would even influence the method, if we had a perfect experimental setup. The next step would be to study the influence of errors which are not inherent to the method. In particular one of the most interesting aspects to be studied would be a non-perfect probe state, i.e. a non-perfect Schrodinger cat-state. ¨ 4 With number of measurements we always mean the number of different probe states used in the experiment.

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

164

Fig. 2. Reconstructed signal density matrix rnŽ1.m of the coherent superposition, Eq. Ž26.. In the left column the real parts of the matrix elements are plotted, whereas the right column contains the corresponding imaginary parts. All the plots have been cut along the diagonal of the density matrix, because the elements on the other side of the diagonal are given by the relation rn m s rm)n . In the top row Ža. the exact elements of rnŽ1.m are shown. The coherence of the state is represented by the appearance of strong non-diagonal elements. We compare the exact elements to the reconstructed elements in the next two rows. Here we have chosen the radius r s 2 for the two coherent components of the probe mode. The middle Žb. and bottom Žc. row show the matrix elements that have been reconstructed by performing M = M s 30 = 30 and M = M s 40 = 40 measurements, respectively. Matrix elements inside a region with n,m - Mr2 are reconstructed very well. Outside this area the behavior of the reconstruction scheme is unpredictable and in general the matrix elements cannot be determined anymore.

equidistant phases f and f X , respectively. From this grid of probabilities the elements rnm are reconstructed with the help of Eqs. Ž22. – Ž24. and via a discretized version of Eq. Ž21., i.e. we replace the Fourier integrals by finite sums according to p

df

p

d fX

X

M

1

M

Hyp 2p Hy p 2p exp w i Ž k f y lf . x ™ M Ý Ý exp 2

ts1 us1

ž

i

2p M

/

Ž kt y lu . .

Ž 28 .

The reconstructed elements of the coherent superposition rˆ Ž1. are shown in Figs. 2Žb. and 2Žc. for M s 30 and M s 40, respectively. In addition we have chosen the radius r s 2 for the reconstruction. We find a faithful reconstruction for Ž1. Ž Mr2. = Ž Mr2. elements of the Fock representation rnm . Above that limit the method breaks down and we can no longer

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

165

Fig. 3. Reconstructed signal density matrix rnŽ2.m of the incoherent superposition, Eq. Ž27.. The structure of this figure is equivalent to Fig. 2. The top row Ža. shows the exact elements, whereas reconstructed elements are shown in the other two rows for M = M s 30 = 30 Žb. and M = M s 40 = 40 Žc.. Again we have chosen r s 2. In contrast to rnŽ1.m the density matrix rnŽ2.m does not possess non-zero elements far off the diagonal. This demonstrates the lack of coherence in state rˆ Ž2.. As for rnŽ1.m the reconstruction scheme works very well inside a region bounded by n,m - Mr2 but breaks down outside this region.

rely on the reconstructed quantities. However, we emphasize that the reconstruction below that limit shows all the features of the exact values plotted in Fig. 2Ža.. In particular, the coherence encoded in the non-diagonal parts can be seen clearly. The reconstruction of the incoherent superposition rˆ Ž2. visualized in Figs. 3Žb. and 3Žc. shows the same behavior. We therefore note that the constraint on the number of reconstructable matrix elements is caused by the discretization of the Fourier integrals, Eq. Ž21.. Such a constraint is well known from the theory of Fourier transforms w28x. The maximal number Ž Mr2. = Ž Mr2. corresponds to the Nyquist condition or Nyquist critical frequency and the phenomenon of getting wrong values outside this limited region is called aliasing. Hence, one of the basic limitations of our method has a well-understood counterpart in classical sampling theory w28x. However, we emphasize that in our particular case we are not probing a classical signal but quantum probabilities and the sampling is done in quantum mechanical phase space spanned by coherent states. The second important parameter that characterizes the quality of the method is the radius r of the reconstruction circle. At first sight this is not so obvious since – looking at Eqs. Ž21. – Ž24. – the method is independent of r. In principle any r

166

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

X

Fig. 4. Visualization of the real 2p-periodic interference terms I Ž re i f ,re i f ., Eq. Ž17., on a radius r s 2. In Ža. we show the interference term I Ž1., which is the complete representation of rˆ Ž1., Eq. Ž26.. The location of the two components of rˆ Ž1. in the complex a-plane can be extracted easily from the plot. We also note the characteristic oscillations visualizing the coherence of rˆ Ž1.. No such oscillations can be seen in the interference term I Ž2. plotted in Žb. for the incoherent superposition rˆ Ž2., Eq. Ž27.. However, the location of the components of the state in the a-plane is visible in this case as well. We emphasize that the phase representation given by the interference term I Ž i. actually serves as an alternative representation of the state rˆ Ž i.. Hence it is tempting to say that it is the complementary representation to the Fock representation shown in Figs. 2Ža. and 3Ža..

can be chosen and we just have to keep it constant in the course of the measurements. However, as soon as we have a discretization of the phases, as is always true in a real experiment, we can strongly improve the quality of the reconstruction with a suitable choice of r. We demonstrate this feature of the method with the help of Fig. 5. It shows the reconstructed elements for the coherent superposition rˆ Ž1. using a phase discretization of M s 40. However, in contrast to Fig. 2Žc., where r s 2 has been chosen, we use now r s 1 in Fig. 5Ža. and r s 5 in Fig. 5Žb.. A comparison of these plots shows that r s 2 is the proper choice for a Ž1. . The choice r s 1 still gives good results for Fock elements of rˆ Ž1. which are close to the nice overall reconstruction rnm origin in the Ž n,m. plane. Or, in other words, a small radius r basically allows us to probe the low-energy contributions to the density operator. On the other hand, the choice r s 5 leads to a reasonable reconstruction of high-energy contributions. The reason why r s 2 works so well in our example is that it probes rˆ Ž1., Eq. Ž26., just in the middle of the contributing coherent amplitudes < g 1 < s 1 and < g 2 < s 3. In principle, this strong influence of r on the reconstruction quality becomes less important when we choose a finer resolution, i.e. when we choose a bigger M. In any real application of the method and without any prior information on rˆs

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

167

Fig. 5. The dependence of the reconstructed density matrix elements rnŽ1.m on the measurement radius r. As in Fig. 2, the real parts of the reconstructed matrix elements are plotted on the left and the imaginary parts on the right. In addition to that we have chosen the number of measurements to be M = M s 40 = 40 so that the results can be directly compared with Fig. 2Žc.. In Ža. a small radius r s 1 was used for the measurements. The density matrix elements for n,m F 10 are still reconstructed well, but the reconstruction breaks down for larger values n,m. The opposite is true, if we use a large radius r s 5 as shown in Žb.. In this case the matrix elements for small n and m cannot be reconstructed properly, whereas the reconstruction works well for higher values of n and m.

one has to play with the parameters M and r in order to see how the aliasing effect vanishes and therefore the quality of the reconstruction improves. But knowing for example the photon statistics rnn , which is a reasonably good prior information, we can concentrate on reconstructing the non-diagonal elements rnm with a radius that should roughly lie in the middle of the distribution rnn as in our examples shown in Figs. 2 and 3.

6. Conclusion We have discussed a reconstruction method for quantum states that describe the signal mode of a heterodyne set-up. At the very heart of this method lies the effect of quantum interference between the state of the signal mode and a properly chosen reference state describing the probe mode. We have shown that a coherent superposition of two coherent states, that is a Schrodinger cat, can serve as a good candidate for this reference state. Operating the heterodyne set-up with such a ¨ well-prepared probe mode leads to interference in the output photocurrent. It is exactly this interference that contains the complete information on the state of the signal mode and we have emphasized that the interference term acts like a phase representation of the signal-mode state. In addition, analyzing the interference is really straightforward since we can uncover the encoded quantum information by a simple Fourier analysis. This simplicity and beauty of the method relies, however, on the ability to prepare a highly

D.G. Fischer, M. Freybergerr Optics Communications 159 (1999) 158–168

168

non-classical state like a Schrodinger cat in the probe mode. This is certainly a drawback to the proposed idea seen in the ¨ light of current technology in quantum optics. Nevertheless, methods for the engineering of coherent state superpositions exist w25,26x and it is not out of reach to have them as a standard tool in a quantum optician’s laboratory in the near future.

Acknowledgements It is a pleasure for us to thank Thomas Beth, Michael Hall, Wolfgang Schleich and Stefan Kienle for many valuable discussions.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x

w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x

J.v. Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955. See the reprint: W. Pauli, General Principles of Quantum Mechanics, Springer, Berlin, 1980. U. Leonhardt, H. Paul, Prog. Quantum Electron. 19 Ž1995. 89. D.-G. Welsch, W. Vogel, T. Opatrny, ´ Homodyne Detection and Quantum State Reconstruction, to appear in Progress in Optics XXXIX, North-Holland, Amsterdam. W.P. Schleich, M.G. Raymer ŽEds.., Special issue on Quantum State Preparation and Measurement, J. Mod. Optics 44 Ž11r12. Ž1997.. K. Vogel, H. Risken, Phys. Rev. A 40 Ž1989. 2847; J. Bertrand, P. Bertrand, Found. Phys. 17 Ž1987. 397. See also U. Leonhardt, Measuring the Quantum State of Light, Cambridge University Press, 1997 and references therein. D.T. Smithey, M. Beck, M.G. Raymer, A. Faridani, Phys. Rev. Lett. 70 Ž1993. 1244. S. Schiller, G. Breitenbach, S.F. Pereira, T. Muller, J. Mlynek, Phys. Rev. Lett. 77 Ž1996. 2933; G. Breitenbach, S. Schiller, J. Mlynek, ¨ Nature 387 Ž1997. 471; G. Breitenbach, S. Schiller, J. Mod. Optics 44 Ž1997. 2207. N.G. Walker, J.E. Carroll, Electron. Lett. 20 Ž1984. 981. T.J. Dunn, J.N. Sweester, I.A. Walmsley, C. Radzewicz, Phys. Rev. Lett. 70 Ž1993. 3388; T.J. Dunn, I.A. Walmsley, S. Mukamel, Phys. Rev. Lett. 74 Ž1995. 884. Ch. Kurtsiefer, T. Pfau, J. Mlynek, Nature 386 Ž1997. 150; T. Pfau, Ch. Kurtsiefer, J. Mod. Optics 44 Ž1997. 2551. D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 77 Ž1996. 4281; D. Leibfried, D.M. Meekhof, C. Monroe, B.E. King, W.M. Itano, D.J. Wineland, J. Mod. Optics 44 Ž1997. 2485. D. Leibfried, T. Pfau, C. Monroe, Physics Today 51 Ž1998. 22. M. Freyberger, P. Bardroff, C. Leichtle, G. Schrade, W.P. Schleich, Physics World 10 Ž1997. 41. G.M. D’Ariano, M.G.A. Paris, Phys. Rev. A 49 Ž1994. 3022. C.W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 1976; A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982; A. Peres, Quantum Theory: Concepts and Methods, Kluwer Academic Press, Dordrecht, 1993. G.M. D’Ariano, in: T. Hakioglu, ˇ A.S. Shumovsky ŽEds.., Quantum Optics and the Spectroscopy of Solids, Kluwer Academic Publisher, Dordrecht, 1997, pp. 139–174. H.P. Yuen, J.H. Shapiro, IEEE Trans. Inf. Theory 24 Ž1978. 657; 25 Ž1979. 179; 26 Ž1980. 78; C.M. Caves, P.D. Drummond, Rev. Mod. Phys. 66 Ž1994. 481. M.J.W. Hall, I.G. Fuss, Quantum Opt. 3 Ž1991. 147; M.J.W. Hall, Phys. Rev. A 50 Ž1994. 3295. V. Bargmann, Commun. Pure Appl. Math. 14 Ž1061. 187. A. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin, 1986. P. Adam, I. Foldesi, J. Janszky, Phys. Rev. A 49 Ž1994. 1281; P. Domokos, P. Adam, J. Janszky, Phys. Rev. A 50 Ž1994. 4293; S. ¨ Szabo, ´ P. Domokos, P. Adam, J. Janszky, Phys. Lett. A 241 Ž1998. 203. W.P. Schleich, M. Pernigo, F. Le Kien, Phys. Rev. A 44 Ž1991. 2172. M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 77 Ž1996. 4887; J.M. Raimond, M. Brune, S. Haroche, Phys. Rev. Lett. 79 Ž1997. 1964. For a review on non-classical states see V. Buzek, ˇ P.L. Knight, in: Progress in Optics XXXIV, North-Holland, Amsterdam, 1995. M. Freyberger, Phys. Rev. A 55 Ž1997. 4120. H. Dym, H.P. McKean, Fourier Series and Integrals, Academic Press, New York, 1972; A. Terras, Harmonic Analysis on Symmetric Spaces and Applications I, Springer-Verlag, New York, 1985; F. Natterer, The Mathematics of Computerized Tomography, Wiley, B.G. Teubner, 1986.