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On measuring extremely small phase fluctuations
Optics Communications 105 (1994) 335-340 North-Holland
OPTICS COMMUNICATIONS
On measuring extremely small phase fluctuations J o h n A. Vaccaro Facultyof CAD, Griffith University,Nathan, Brisbane, 4111Australia and D.T. Pegg Faculty ofScience and Technology, Grifl~th University,Nathan, Brisbane, 4111Australia Received 15 July 1993; revised manuscript received 7 October 1993
The applicabilityis examined of the measured-phaseoperator describinga simple homodyne measurement schemeto states of the quantum optical field which have the minimum possible phase fluctuations. Such states necessarilyhave huge intensity fluctuations, so it is expected that this operator should be inapplicable because intensity fluctuations are deliberately ignored in its derivation. However, the surprising result found is that this operator can still provide a reasonably good approximation to the phase properties of fields even in this extreme limit.
The development of the unitary and hermitian phase operator formalism [ 1-3] has allowed the properties of phase, which is conjugate to photon number, to be successfully calculated for quantum states of light. A question of increasing interest concerns how these properties can be measured [ 4-6 ]. The operationally defined measured-phase operators c - ~ 0 and Sl-'~M0 were introduced as the description of phase-like quantities measurable by a simple balanced homodyne experiment [ 4 ] in which the local oscillator is in a sufficiently intense coherent state to have negligible fluctuations. Other such operators have also been constructed to describe more sophisticated experiments [5,6]. Such operators do not represent a quantity which is conjugate to photon number, that is phase, but do yield quite good approximations to the phase properties of some states of light. The usefulness of measured-phase operators is determined by the range of states for which these approximations are reasonably accurate. Recently, for example, Ritze [ 6 ] has suggested that some measured-phase operators [ 4,5 ], while giving good approximations for coherent states, are not useful for phase optimized states [ 7,8 ]. These states have the minimum possible phase fluctuations possible for a
given mean photon number. In this paper we examine this question more closely and find, remarkably, that the simple measured-phase operator ofref. [ 4 ] can still yield a quite good approximation to actual phase even in the extreme limit of small phase fluctuations. Some results derivable from the hermitian phase operator approach [ 1-3 ] which are used in this paper include expressions for the phase probability densities P,(O), Psin,(0) and Pst~(sin0) and for phase shifts A0 for a physical state of the field. Here P,(O) dO is the probability for the phase ¢ to have a value between 0 and 0+d0; Psi,,(0)dO is the probability for sin0 to have a value between sin0 and s i n ( 0 + d 0 ) ; and P~in~(sin0) d(sin0) is the probability for sin0 to have a value between sin0 and sin0 + d (sin0). In terms of number state coefficients ( n I f ) and phase state coefficients ( 0 I f ) , these are, for a physical state I f ) [ 1-3,9 ] s+l P,(O)= lim-~-I
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All fights r e s e r v e d . SSDI 0030-401 8 (93)E0549-U
(Oil)
[2
(1)
x~oo 1
oo
-~n,~,exp[t(n'-n)Ol
(fin') (nlf) ,
(2) 335
Volume 105, number 5,6
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lim s + 1 (l(OIf) 12+l(n-OIf)12) esino(O)= s~o~-2"~-n
(3)
1 oo =}-nn,~, {exp[ i( n ' - n )O]
+ ( - 1 )n'-n e x p [ - i ( n ' - n ) O ] }
Psin~(sin0) = Psin~(0)/cos0.
, (4) (5)
In eqs. ( 1 ) and (3), s + 1 is the dimensionality of the Hilbert space spanned by the orthogonal phase states I 0m>, with rn = 0, 1, ..., s, which are a subset of all the phase states 10) = (1-I-s) -1/2 Zexp(in0) In> where the sum is over n = 0, 1, ..., s [ 1 ]. In eqs. ( 3 ) (5) 0 is between - n/2 and n/2 and sin0 ranges from - 1 to + 1. The extra term in eq. (3) compared with eq. (1) results from the two-fold degeneracy of the operator sin~o, of which both [0) and I n - 0 ) are eigenstates with eigenvalue sin0. A shift A0 in phase of a quantum state is produced by the unitary phase-shift operator exp (i~A0) where ~r is the number operator. It is easily checked that application of this operator to I f ) translates the probability density ( 1 ) along the 0 axis by A0. How can we produce a phase shift experimentally? Writing ~rA0=~mAt, where At=AO/to and ~ho9 is the field hamiltonian g , and remembering that ,rf/h is the generator of a time shift, we see that a phase shift can be produced by inserting an effective time delay. This is, in fact, the usual procedure in most experiments, which leads to the important result that the operational phase shift operator is identical to the actual phase shift operator. The measured-phase operators are hermitian combinations of the creation and annihilation operators c-"ff~MO=k(d+~t),
(6)
s~-'~M~ = - i k ( • - a t ) ,
(7)
where k is a state-dependent factor which must approach (2 ( ~ ) 1/2 ) - 1 for an intense coherent state to give the correct classical limit [4 ]. A useful compromise value applicable to a wide range of states was found to be k = 2 - ~( ( N ) + 1 / 2 ) - 1/2. Clearly in eqs. (6) and (7), 2kfi is used as a quasi-unitary phase operator in place of the unitary phase operator exp (i~o). From the relation a = exp (i~o)/(r ~/2 [ 1 ] we see that ~ / 2 is being treated as a state-dependent 336
15 February 1994
constant which ignores its operator nature. However, because n -1/2 f~ I n ) = e x p ( i ~ o ) I n ) for n > 0 , this approximation will be reasonable for states with an uncertainty in photon number which is small compared with the mean photon number. This is the case for number states or coherent states of reasonable intensity. We shall use the term Airy function state, or Airy state, of light to refer to the state whose number state coefficients fit an Airy function curve as described in ref. [7 ] as the state with minimum phase uncertainty for a given mean photon number (~r) of the order of two and higher. These phase-optimized states have also been studied in ref. [ 8 ] and their possible production in ref, [ 10 ]. The phase variance is given by A02~. 1.88 ( ( ~ ) +0.86) -2. Thus for a coherent state to have the same phase uncertainty as an Airy state with mean photon number of 50 photons, for example, it must have a mean of over 300 photons. As pointed out by Ritze [ 6 ], these extremely small phase fluctuations imply, from the number-phase uncertainty relation [2,3] that the number uncertainty AN must be at least of the order (~r). This suggests that the above measured-phase operators, and some other measured-phase operators [5 ] will not in general represent good measurements of the small phase variance of an Airy state. I f an Airy state is to be used successfully to convey a similar amount of phase information as a coherent state of much greater intensity, then a detection experiment capable of resolving the minimum phase uncertainty is essential. In view of the simplicity of the homodyne measurements represented by the measured-phase operators, it is worthwhile examining the problem more closely. The balanced homodyne experiment incorporates a 50%: 50% beam splitter with the signal being the electronically obtained difference between the outputs from two photodetectors [5,6]. The phase of the input field is the phase difference between it and a local oscillator field whose phase we define to be n/2. In the classical case the minimum signal magnitude of zero occurs when the input field is in quadrature with the local field, which will be the case when the phase of the input is zero. With this phase the contribution to the signal from intensity fluctuations in the input field is also minimized. A fluctuation from zero of 0 in the input field phase, however, pro-
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duces a signal proportional to sin0, which for small fluctuations is approximately 0. Thus if we adjust the phase of the input field until the mean signal is zero then, for small fluctuations, the variance in phase will be ( 02) ~ (sin20) which, providing we can ignore the effect of intensity fluctuations, is obtainable from the fluctuations in the signal. In the corresponding quantum mechanical case, assuming a suitable photon counting time interval [ 5,6 ], the operator representing the signal is B Sl-q'n~MO,where B is a proportionality constant, which allows us to find ((sl-Tfi'~MO)2 ) as the measured-phase variance. (We note that in ref. [4] the signal is proportional to 0- This is because the local oscillator phase [4] was defined as being zero instead of - n / 2 as it is here.) In table 1 we give the values of the standard deviation ((SI-~M~) 2 ) 1/2 in measured phase calculated with the value of k given above for Airy states with mean photon numbers of 5, 10 and 100. These are compared directly with the values of standard deviation ((sin~o)2> 1/2 obtained from the hermitian phase operator for the same states and the corresponding values of the standard deviation (~2 ~ ~/2 in phase. The agreement is certainly not as good as for coherent states. For example, the corresponding values of ( (SI-~M~)2 ~ 1/2 and ( (sin~o)2 ~ i/2 for a coherent state with a mean photon number of 10 are 0.154 and 0.161, respectively, and for a mean photon number of 100 are 0.0499 and 0.05006, respectively. (The corresponding values of (~2 ~ ~/2 for means of 10 and 100 photons are 0.163 and 0.05013. ) Nevertheless the disagreement as shown in table 1 is not so great as to prevent us from using the measured-phase operator to give a good indication of the actual phase uncertainty. Also, the advantage that the Airy states have in phase resolution as given by the standard deviation is not seriously affected if detected by this means.
15 February 1994
In the above calculations we used the compromise value for the factor k given earlier. Although the value of k could be determined by a separate measurement of ( N ) , a simpler method in practice to measure the standard deviation of phase would be as follows. The possible values S of the signal are divided into a large number of very small bins of size 8S. The phase, either of the local oscillator or the input field, is shifter until the probability for the signal to be in the zeroS bin is maximized. Airy states, like other partial phase states [ 3], have a symmetric phase distribution with the maximum coinciding with the mean. Also the maxima in the measured-phase distribution and in the actual phase distribution coincide when the mean phase is zero. Thus the phase shift needed to maximize the zero-S bin signal is equal to the actual original mean phase. This is because the operational phase shift is equal to the actual phase shift. With the phase adjusted so that the mean signal value is zero, the probability Pr(S) for the signal in each bin in measured. The phase is then shifted through an extra amount A0 and the shift A ( S ) of the mean of the distribution, with standard deviation as, is found. The standard deviation of the measured phase, that is aJB, is then taken as the measured result as AO/(A(S) ). We can show that this procedure is equivalent, for small A0, to using a value for k in eq.
(7) of k = [ (fo I (~+~*)Ifo ~ l -~ ,
(8)
where here Ifo) is the input state after its mean phase has been shifted to zero. Expression (8) follows from equating aJB to trs A 0 / ( A ( S ) ) for small A0 which is equivalent to setting d(Sl-'i~M~ ) / d O to unity. This last expression is equal to - i ( [~r, s q i ~ ] ) because is the generator of a phase shift. Substitution from eq. (7) then yields eq. (8). We find that the measured-phase standard deviation based on the alter-
Table 1 Comparison of standard deviations of phase for an Airy state with various values of mean photon number. Columns 2-5 contain the standard deviations of the measured sine operator, the sine of the hermitian phase operator, the hermitian phase operator and the distribution QM(0), respectively. (~)
( (SX'~M~) 2) ,/2
((sin~a) 2>,/2
(~ >
aO
5 10 100
0.183 0.100 0.011
0.219 0.123 0.0136
0.234 0.126 0.0136
0.230 0.132 0.0156 337
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native operational value o f k given by eq. (8) are not appreciably different from those based on the compromise value of k for the Airy states with the same mean photon numbers given earlier. The variance represents a combination of only the first and second moments of the probability distribution, and is not the only measure of the phase resolution. Therefore the following question immediately arises. In view of the only approximate agreement in the measured-phase and actual phase variances, how good an agreement is there between measured and actual phase distribution? From the experimental results for the probability density Pr(S)/SS as a function of S, we can obtain a probability density as a function of sin0 by using the scaling factor A0/(AS) discussed previously. From our comments above, the distribution obtained from this scaling factor is not significantly different from the distribution obtained using the compromise value of k, which we call the measured-phase probability density PM(sin0). We can calculate PM(sin0) for a state I f ) as follows. Let [y> be the eigenstate of - i ( f i - f i * ) with eigenvalue y, that is, sl-~M01y)= k y l y ) . This state has number state coefficients
15 February 1994
0 -0.7
40
0 sin(O)
0.7
I 30
20
12. Finally the probability density is normalized. A comparison of the predicted measured-phase probability density PM(sin0) (dashed line) with the corresponding actual phase probability density, which is P, in,(sin0) (solid line), for Airy states with (N> = 5, 10 and 100 is shown in fig. 1. It can be seen that the measuredphase distributions are not unreasonable representations of the actual phase distributions. We note from eqs. ( 1 ) and (3) that for states such as these, for which (f[ ~ - 0 ) is negligible, Psin~(sin0) is proportional to P,( O)/cos0, which for small phase variances about zero is a approximately P,(O). In order to find the mean phase by the above method, the values of the zero-S bin probability P r ( 0 ) need to be monitored as the phase of the field is shifted. In view of this, another possible measured-phase distribution immediately suggests itself: simply plot these values of Pr (0) as a function of the 338
-0.05
0 0.05 sin(O)
Fig. 1. Comparisonsof the predicted measured-phaseprobability distributions PM(sin0) (dashed lines) with the corresponding actual phase probabilitydistributions Pine(sin0) (solid lines) for Airy states with mean photon numbers of (a) 5 and 10 and (b) 100. Note that in (b) the horizontal scale is expanded. phase shift 0 (see also ref. [ 12 ] ). These values will be proportional to PM (0) for the phase shifted state exp(-ii~Tt) Lf) = If(O)> and thus, from eq. (9), will be proportional to
I (f(O)ly=O> 12 2
oc ~i"2-("/2)(n!)-l/2Hn(O) exp(in0) , n=O
(lO)
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15 February 1994
because the number state coefficients are now =
4
s+l Q ( 0 ) = l i m - ~ -n- (l(flO) 12+l(fln+O)12),
0 /4
(11)
$ ~ o o
where we have used exp(ifi?0) Izero p h a s e ) = 10) and exp(iN0) I n ) = I n + 0 ) . For states I f ) with negligible probability of being found in I n + 0>, the latter term in eq. ( 11 ) vanishes and, from eq. ( 1 ), we obtain
Q(O)=Po(O) .
I
2
40
0 0 (
/t/4
,
(12)
The operational procedure which yields QM (0) has the advantage that it is independent of the value used for k and there is no need to use the scaling constant A0/(AS). Figure 2 shows a comparison between Qra(O) (dotted line) and P~(O) (solid line) for the same states as in fig. 1 over a range near the mean. We might note at this point that the measured-phase operator of Noh et el. [ 5 ] also has the advantage of not requiring a scaling constant or k value, but as the results it yields are a function of four input states [ 13 ] they will not be identical to those given here. It is clear that the operational procedure giving the measured-phase probability density QM (0) yields a better measure of the actual phase probability density than that which gives PM(sin0). Indeed the agreement in the former case is reasonably good. The standard deviations trQ of QM(O) for the same Airy states considered previously are shown in the last column of table 1. Thus the simple balanced homodyne experiment, if used in the appropriate manner, can give a measured-phase distribution which is a reasonably accurate representation of the actual phase distribution even for Airy states. This is important because, although the phase distribution of any state can be determined by measuring the density matrix [ 14-16 ], the above methods are much simpler and far more direct. It is also very surprising
20
_
-rd40
__
0 0
rt/40
Fig. 2. Comparisons between the distributions QM(O) (dotted
lines) and the actual phase probabilitydistributions P~(0) (solid lines) for Airy states with mean photon numbers of (a) 5 and 10 and (b) 100. since, as pointed out by Ritze [6 ], the validity of approximating the actual phase operator by the measured-phase operator is expected to break down for such extreme states. Indeed, because Airy states with a mean of more than a few photons are phase-optimized states, it is impossible to find more extreme states in this particular limit of extremely small phase variances. That is, as we move from coherent states to states of decreasing phase variance while keeping the mean photon number fixed, the simple measured-phase operator becomes less accurate, but we reach the limit of such states before we reach the limit 339
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o f reasonably g o o d applicability for this operator. T h e w o r k in this p a p e r thus c o n f i r m s t h e v a l i d i t y o f the s i m p l e m e a s u r e d - p h a s e o p e r a t o r o v e r a v e r y useful range o f states w i t h a m e a n f r o m a few p h o t o n s up to at least h u n d r e d s o f p h o t o n s . A p p l i c a t i o n s o f this o p e r a t o r by m e a n s o f t h e t w o d i f f e r e n t e x p e r i m e n t a l p r o c e d u r e s d e s c r i b e d a b o v e to states in o t h e r regions, such as that o f v e r y small p h o t o n n u m b e r , will be e x a m i n e d elsewhere.
References [1 ] D.T. Pegg and S.M. Barnett, Europhys, Len. 6 (1988 ) 483. [2] S.M. Barnett and D.T. Pegg, J. Mod. Optics 36 (1989) 7. [3] D.T. Pegg and S.M. Barnett, Phys. Rev. A 39 (1989) 1665.
340
15 February 1994
[4] S.M. Barnett and D.T. Pegg, J. Phys. A 19 (1986) 3849. [5] J.W. Noh, A. Fougeres and L. Mandel, Phys. Rev. A 45 ( 1992 ) 424. [6] H.-H. Ritze, Optics Comm. 92 (1992) 127. [7] G.S. Summy and D.T. Pegg, Optics Comm. 77 (1990) 75. [8 ] A. Bandilla, H. Paul and H.-H. Ritz¢, Quantum Optics 3 (1991) 267. [9] J.A. Vaccaro and D.T. Pegg, J. Mod. Optics 37 (1990) 17. [10] A. Bandilla, Optics Comm. 94 (1992 ) 273. [ 11 ] E. Merzbacher, Quantum Mechanics, 2nd Ed. (Wiley, New York, 1970) p. 362. [ 12 ] W. Vogel and W. Schleich, Phys. Rev. A 44 ( 1991 ) 7642. [ 13 ] M. Freyberger, K. Vogel and W. Schleich, Phys. Lett. A 176 (1993) 41. [ 14 ] D.T. Smithey, M. Beck, M.G. Raymer and A. Faridani, Phys. Rev. Lett. 70 (1993) 1244. [ 15 ] K. Vogel and H. Risken, Phys. Rev. A 40 (1989) 2847. [ 16] M. Beck, D.T. Smithey, J. Cooper and M.G. Raymer, Optics Lett. 18 (1993) 1259.
OPTICS COMMUNICATIONS
On measuring extremely small phase fluctuations J o h n A. Vaccaro Facultyof CAD, Griffith University,Nathan, Brisbane, 4111Australia and D.T. Pegg Faculty ofScience and Technology, Grifl~th University,Nathan, Brisbane, 4111Australia Received 15 July 1993; revised manuscript received 7 October 1993
The applicabilityis examined of the measured-phaseoperator describinga simple homodyne measurement schemeto states of the quantum optical field which have the minimum possible phase fluctuations. Such states necessarilyhave huge intensity fluctuations, so it is expected that this operator should be inapplicable because intensity fluctuations are deliberately ignored in its derivation. However, the surprising result found is that this operator can still provide a reasonably good approximation to the phase properties of fields even in this extreme limit.
The development of the unitary and hermitian phase operator formalism [ 1-3] has allowed the properties of phase, which is conjugate to photon number, to be successfully calculated for quantum states of light. A question of increasing interest concerns how these properties can be measured [ 4-6 ]. The operationally defined measured-phase operators c - ~ 0 and Sl-'~M0 were introduced as the description of phase-like quantities measurable by a simple balanced homodyne experiment [ 4 ] in which the local oscillator is in a sufficiently intense coherent state to have negligible fluctuations. Other such operators have also been constructed to describe more sophisticated experiments [5,6]. Such operators do not represent a quantity which is conjugate to photon number, that is phase, but do yield quite good approximations to the phase properties of some states of light. The usefulness of measured-phase operators is determined by the range of states for which these approximations are reasonably accurate. Recently, for example, Ritze [ 6 ] has suggested that some measured-phase operators [ 4,5 ], while giving good approximations for coherent states, are not useful for phase optimized states [ 7,8 ]. These states have the minimum possible phase fluctuations possible for a
given mean photon number. In this paper we examine this question more closely and find, remarkably, that the simple measured-phase operator ofref. [ 4 ] can still yield a quite good approximation to actual phase even in the extreme limit of small phase fluctuations. Some results derivable from the hermitian phase operator approach [ 1-3 ] which are used in this paper include expressions for the phase probability densities P,(O), Psin,(0) and Pst~(sin0) and for phase shifts A0 for a physical state of the field. Here P,(O) dO is the probability for the phase ¢ to have a value between 0 and 0+d0; Psi,,(0)dO is the probability for sin0 to have a value between sin0 and s i n ( 0 + d 0 ) ; and P~in~(sin0) d(sin0) is the probability for sin0 to have a value between sin0 and sin0 + d (sin0). In terms of number state coefficients ( n I f ) and phase state coefficients ( 0 I f ) , these are, for a physical state I f ) [ 1-3,9 ] s+l P,(O)= lim-~-I
0030-4018/94/$07.00 © 1994 Elsevier Science B.V. All fights r e s e r v e d . SSDI 0030-401 8 (93)E0549-U
(Oil)
[2
(1)
x~oo 1
oo
-~n,~,exp[t(n'-n)Ol
(fin') (nlf) ,
(2) 335
Volume 105, number 5,6
OPTICS COMMUNICATIONS
lim s + 1 (l(OIf) 12+l(n-OIf)12) esino(O)= s~o~-2"~-n
(3)
1 oo =}-nn,~, {exp[ i( n ' - n )O]
+ ( - 1 )n'-n e x p [ - i ( n ' - n ) O ] }
Psin~(sin0) = Psin~(0)/cos0.
In eqs. ( 1 ) and (3), s + 1 is the dimensionality of the Hilbert space spanned by the orthogonal phase states I 0m>, with rn = 0, 1, ..., s, which are a subset of all the phase states 10) = (1-I-s) -1/2 Zexp(in0) In> where the sum is over n = 0, 1, ..., s [ 1 ]. In eqs. ( 3 ) (5) 0 is between - n/2 and n/2 and sin0 ranges from - 1 to + 1. The extra term in eq. (3) compared with eq. (1) results from the two-fold degeneracy of the operator sin~o, of which both [0) and I n - 0 ) are eigenstates with eigenvalue sin0. A shift A0 in phase of a quantum state is produced by the unitary phase-shift operator exp (i~A0) where ~r is the number operator. It is easily checked that application of this operator to I f ) translates the probability density ( 1 ) along the 0 axis by A0. How can we produce a phase shift experimentally? Writing ~rA0=~mAt, where At=AO/to and ~ho9 is the field hamiltonian g , and remembering that ,rf/h is the generator of a time shift, we see that a phase shift can be produced by inserting an effective time delay. This is, in fact, the usual procedure in most experiments, which leads to the important result that the operational phase shift operator is identical to the actual phase shift operator. The measured-phase operators are hermitian combinations of the creation and annihilation operators c-"ff~MO=k(d+~t),
(6)
s~-'~M~ = - i k ( • - a t ) ,
(7)
where k is a state-dependent factor which must approach (2 ( ~ ) 1/2 ) - 1 for an intense coherent state to give the correct classical limit [4 ]. A useful compromise value applicable to a wide range of states was found to be k = 2 - ~( ( N ) + 1 / 2 ) - 1/2. Clearly in eqs. (6) and (7), 2kfi is used as a quasi-unitary phase operator in place of the unitary phase operator exp (i~o). From the relation a = exp (i~o)/(r ~/2 [ 1 ] we see that ~ / 2 is being treated as a state-dependent 336
15 February 1994
constant which ignores its operator nature. However, because n -1/2 f~ I n ) = e x p ( i ~ o ) I n ) for n > 0 , this approximation will be reasonable for states with an uncertainty in photon number which is small compared with the mean photon number. This is the case for number states or coherent states of reasonable intensity. We shall use the term Airy function state, or Airy state, of light to refer to the state whose number state coefficients fit an Airy function curve as described in ref. [7 ] as the state with minimum phase uncertainty for a given mean photon number (~r) of the order of two and higher. These phase-optimized states have also been studied in ref. [ 8 ] and their possible production in ref, [ 10 ]. The phase variance is given by A02~. 1.88 ( ( ~ ) +0.86) -2. Thus for a coherent state to have the same phase uncertainty as an Airy state with mean photon number of 50 photons, for example, it must have a mean of over 300 photons. As pointed out by Ritze [ 6 ], these extremely small phase fluctuations imply, from the number-phase uncertainty relation [2,3] that the number uncertainty AN must be at least of the order (~r). This suggests that the above measured-phase operators, and some other measured-phase operators [5 ] will not in general represent good measurements of the small phase variance of an Airy state. I f an Airy state is to be used successfully to convey a similar amount of phase information as a coherent state of much greater intensity, then a detection experiment capable of resolving the minimum phase uncertainty is essential. In view of the simplicity of the homodyne measurements represented by the measured-phase operators, it is worthwhile examining the problem more closely. The balanced homodyne experiment incorporates a 50%: 50% beam splitter with the signal being the electronically obtained difference between the outputs from two photodetectors [5,6]. The phase of the input field is the phase difference between it and a local oscillator field whose phase we define to be n/2. In the classical case the minimum signal magnitude of zero occurs when the input field is in quadrature with the local field, which will be the case when the phase of the input is zero. With this phase the contribution to the signal from intensity fluctuations in the input field is also minimized. A fluctuation from zero of 0 in the input field phase, however, pro-
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duces a signal proportional to sin0, which for small fluctuations is approximately 0. Thus if we adjust the phase of the input field until the mean signal is zero then, for small fluctuations, the variance in phase will be ( 02) ~ (sin20) which, providing we can ignore the effect of intensity fluctuations, is obtainable from the fluctuations in the signal. In the corresponding quantum mechanical case, assuming a suitable photon counting time interval [ 5,6 ], the operator representing the signal is B Sl-q'n~MO,where B is a proportionality constant, which allows us to find ((sl-Tfi'~MO)2 ) as the measured-phase variance. (We note that in ref. [4] the signal is proportional to 0- This is because the local oscillator phase [4] was defined as being zero instead of - n / 2 as it is here.) In table 1 we give the values of the standard deviation ((SI-~M~) 2 ) 1/2 in measured phase calculated with the value of k given above for Airy states with mean photon numbers of 5, 10 and 100. These are compared directly with the values of standard deviation ((sin~o)2> 1/2 obtained from the hermitian phase operator for the same states and the corresponding values of the standard deviation (~2 ~ ~/2 in phase. The agreement is certainly not as good as for coherent states. For example, the corresponding values of ( (SI-~M~)2 ~ 1/2 and ( (sin~o)2 ~ i/2 for a coherent state with a mean photon number of 10 are 0.154 and 0.161, respectively, and for a mean photon number of 100 are 0.0499 and 0.05006, respectively. (The corresponding values of (~2 ~ ~/2 for means of 10 and 100 photons are 0.163 and 0.05013. ) Nevertheless the disagreement as shown in table 1 is not so great as to prevent us from using the measured-phase operator to give a good indication of the actual phase uncertainty. Also, the advantage that the Airy states have in phase resolution as given by the standard deviation is not seriously affected if detected by this means.
15 February 1994
In the above calculations we used the compromise value for the factor k given earlier. Although the value of k could be determined by a separate measurement of ( N ) , a simpler method in practice to measure the standard deviation of phase would be as follows. The possible values S of the signal are divided into a large number of very small bins of size 8S. The phase, either of the local oscillator or the input field, is shifter until the probability for the signal to be in the zeroS bin is maximized. Airy states, like other partial phase states [ 3], have a symmetric phase distribution with the maximum coinciding with the mean. Also the maxima in the measured-phase distribution and in the actual phase distribution coincide when the mean phase is zero. Thus the phase shift needed to maximize the zero-S bin signal is equal to the actual original mean phase. This is because the operational phase shift is equal to the actual phase shift. With the phase adjusted so that the mean signal value is zero, the probability Pr(S) for the signal in each bin in measured. The phase is then shifted through an extra amount A0 and the shift A ( S ) of the mean of the distribution, with standard deviation as, is found. The standard deviation of the measured phase, that is aJB, is then taken as the measured result as AO/(A(S) ). We can show that this procedure is equivalent, for small A0, to using a value for k in eq.
(7) of k = [ (fo I (~+~*)Ifo ~ l -~ ,
(8)
where here Ifo) is the input state after its mean phase has been shifted to zero. Expression (8) follows from equating aJB to trs A 0 / ( A ( S ) ) for small A0 which is equivalent to setting d(Sl-'i~M~ ) / d O to unity. This last expression is equal to - i ( [~r, s q i ~ ] ) because is the generator of a phase shift. Substitution from eq. (7) then yields eq. (8). We find that the measured-phase standard deviation based on the alter-
Table 1 Comparison of standard deviations of phase for an Airy state with various values of mean photon number. Columns 2-5 contain the standard deviations of the measured sine operator, the sine of the hermitian phase operator, the hermitian phase operator and the distribution QM(0), respectively. (~)
( (SX'~M~) 2) ,/2
((sin~a) 2>,/2
(~ >
aO
5 10 100
0.183 0.100 0.011
0.219 0.123 0.0136
0.234 0.126 0.0136
0.230 0.132 0.0156 337
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native operational value o f k given by eq. (8) are not appreciably different from those based on the compromise value of k for the Airy states with the same mean photon numbers given earlier. The variance represents a combination of only the first and second moments of the probability distribution, and is not the only measure of the phase resolution. Therefore the following question immediately arises. In view of the only approximate agreement in the measured-phase and actual phase variances, how good an agreement is there between measured and actual phase distribution? From the experimental results for the probability density Pr(S)/SS as a function of S, we can obtain a probability density as a function of sin0 by using the scaling factor A0/(AS) discussed previously. From our comments above, the distribution obtained from this scaling factor is not significantly different from the distribution obtained using the compromise value of k, which we call the measured-phase probability density PM(sin0). We can calculate PM(sin0) for a state I f ) as follows. Let [y> be the eigenstate of - i ( f i - f i * ) with eigenvalue y, that is, sl-~M01y)= k y l y ) . This state has number state coefficients
15 February 1994
0 -0.7
40
0 sin(O)
0.7
I 30
20
-0.05
0 0.05 sin(O)
Fig. 1. Comparisonsof the predicted measured-phaseprobability distributions PM(sin0) (dashed lines) with the corresponding actual phase probabilitydistributions Pine(sin0) (solid lines) for Airy states with mean photon numbers of (a) 5 and 10 and (b) 100. Note that in (b) the horizontal scale is expanded. phase shift 0 (see also ref. [ 12 ] ). These values will be proportional to PM (0) for the phase shifted state exp(-ii~Tt) Lf) = If(O)> and thus, from eq. (9), will be proportional to
I (f(O)ly=O> 12 2
oc ~
(lO)
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OPTICS COMMUNICATIONS
15 February 1994
because the number state coefficients
4
s+l Q ( 0 ) = l i m - ~ -n- (l(flO) 12+l(fln+O)12),
0 /4
(11)
$ ~ o o
where we have used exp(ifi?0) Izero p h a s e ) = 10) and exp(iN0) I n ) = I n + 0 ) . For states I f ) with negligible probability of being found in I n + 0>, the latter term in eq. ( 11 ) vanishes and, from eq. ( 1 ), we obtain
Q(O)=Po(O) .
I
2
40
0 0 (
/t/4
,
(12)
The operational procedure which yields QM (0) has the advantage that it is independent of the value used for k and there is no need to use the scaling constant A0/(AS). Figure 2 shows a comparison between Qra(O) (dotted line) and P~(O) (solid line) for the same states as in fig. 1 over a range near the mean. We might note at this point that the measured-phase operator of Noh et el. [ 5 ] also has the advantage of not requiring a scaling constant or k value, but as the results it yields are a function of four input states [ 13 ] they will not be identical to those given here. It is clear that the operational procedure giving the measured-phase probability density QM (0) yields a better measure of the actual phase probability density than that which gives PM(sin0). Indeed the agreement in the former case is reasonably good. The standard deviations trQ of QM(O) for the same Airy states considered previously are shown in the last column of table 1. Thus the simple balanced homodyne experiment, if used in the appropriate manner, can give a measured-phase distribution which is a reasonably accurate representation of the actual phase distribution even for Airy states. This is important because, although the phase distribution of any state can be determined by measuring the density matrix [ 14-16 ], the above methods are much simpler and far more direct. It is also very surprising
20
_
-rd40
__
0 0
rt/40
Fig. 2. Comparisons between the distributions QM(O) (dotted
lines) and the actual phase probabilitydistributions P~(0) (solid lines) for Airy states with mean photon numbers of (a) 5 and 10 and (b) 100. since, as pointed out by Ritze [6 ], the validity of approximating the actual phase operator by the measured-phase operator is expected to break down for such extreme states. Indeed, because Airy states with a mean of more than a few photons are phase-optimized states, it is impossible to find more extreme states in this particular limit of extremely small phase variances. That is, as we move from coherent states to states of decreasing phase variance while keeping the mean photon number fixed, the simple measured-phase operator becomes less accurate, but we reach the limit of such states before we reach the limit 339
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o f reasonably g o o d applicability for this operator. T h e w o r k in this p a p e r thus c o n f i r m s t h e v a l i d i t y o f the s i m p l e m e a s u r e d - p h a s e o p e r a t o r o v e r a v e r y useful range o f states w i t h a m e a n f r o m a few p h o t o n s up to at least h u n d r e d s o f p h o t o n s . A p p l i c a t i o n s o f this o p e r a t o r by m e a n s o f t h e t w o d i f f e r e n t e x p e r i m e n t a l p r o c e d u r e s d e s c r i b e d a b o v e to states in o t h e r regions, such as that o f v e r y small p h o t o n n u m b e r , will be e x a m i n e d elsewhere.
References [1 ] D.T. Pegg and S.M. Barnett, Europhys, Len. 6 (1988 ) 483. [2] S.M. Barnett and D.T. Pegg, J. Mod. Optics 36 (1989) 7. [3] D.T. Pegg and S.M. Barnett, Phys. Rev. A 39 (1989) 1665.
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[4] S.M. Barnett and D.T. Pegg, J. Phys. A 19 (1986) 3849. [5] J.W. Noh, A. Fougeres and L. Mandel, Phys. Rev. A 45 ( 1992 ) 424. [6] H.-H. Ritze, Optics Comm. 92 (1992) 127. [7] G.S. Summy and D.T. Pegg, Optics Comm. 77 (1990) 75. [8 ] A. Bandilla, H. Paul and H.-H. Ritz¢, Quantum Optics 3 (1991) 267. [9] J.A. Vaccaro and D.T. Pegg, J. Mod. Optics 37 (1990) 17. [10] A. Bandilla, Optics Comm. 94 (1992 ) 273. [ 11 ] E. Merzbacher, Quantum Mechanics, 2nd Ed. (Wiley, New York, 1970) p. 362. [ 12 ] W. Vogel and W. Schleich, Phys. Rev. A 44 ( 1991 ) 7642. [ 13 ] M. Freyberger, K. Vogel and W. Schleich, Phys. Lett. A 176 (1993) 41. [ 14 ] D.T. Smithey, M. Beck, M.G. Raymer and A. Faridani, Phys. Rev. Lett. 70 (1993) 1244. [ 15 ] K. Vogel and H. Risken, Phys. Rev. A 40 (1989) 2847. [ 16] M. Beck, D.T. Smithey, J. Cooper and M.G. Raymer, Optics Lett. 18 (1993) 1259.