- Email: [email protected]
Quantum-non-demolition measurement of the phase
12June 1995 PHYSICS LETTERS
EISEVIER
A
Physics Letters A 202 (1995) l-6
Quantum-non-demolition measurement of the phase V.B
. Braginsky ‘, EYa. Khalili, A.A. Kulaga
Chair of Molecular Physics and Physical Measurements, Department of Physics, Moscow State University, 119899 Moscow, Russian Federation
Received 4 March 1995; accepted for publication 17 March 1995 Communicated by V.M. Agranovich
Abstract The problem of the QND measurement of the phase of a linear quantum oscillator is analyzed. It is shown that by measuring the quadrature amplitude one can reach a resolution close to the fundamental limit in a narrow range of phase values. A new method of QND phase measurement is proposed and analyzed. This method is also based on the coordinate interaction between the quantum object and the measuring device but is free from the limitation mentioned above. Several practical schemes of phase measurementsare discussed.
1. Introduction The group of problems associated with the phase observable has been for a long time, and partially remained until now, an incomplete part of quantum theory. The main reason for this is the absence of a “good” operator of the phase defined on the entire space of oscillator states. The analysis of the mathematical aspects of this problem can be found in Ref. [ 11. In the last 20 years, the formal quantum theory of the phase has been substantially developed (see Ref. [ 21). However, presently a complete quantummechanical description of the phase as a physical quantity does not exist. Such a description must necessarily include a definition (at least gedanken) of its measurement procedure. In this area, the progress has not been impressive. Till now, to the authors’ knowledge, not a single (even gedanken) procedure of “true phase” measurement has been proposed, which under
1E-mail: [email protected].
ideal conditions would reduce the wavefunction of the oscillator to a state with a given phase. The realization of such a scheme would allow one, for example, to substantially improve one of the basic methods of spectroscopy - collation of frequencies of two oscillators (one of them can be the basic element in a high-stability self-oscillator). It can be also mentioned that the phase of the harmonic oscillator is a QND observable [ 31 and therefore allows, in principle, for a continuous non-perturbative monitoring. Apparently, it is the single simple example of the QND observables that is not an integral of motion. The difficulty of realization of a “true phase” measurement is that the meter should respond only to the phase of the oscillations and should in no way respond to the amplitude. One can hardly imagine a physical interaction that would meet the requirement above. In this paper, we consider two methods for measuring the oscillator phase. Although based on the usual (coordinate) interaction of the meter with the object under study, they allow one to obtain a precision of the phase measurement close to the theoretical limit.
0375-9601/95/@9.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00279-O
2
V.B. Braginsky et al. /Physics
Letters A 202 (I 995) I-6
2. On the ultimate precision in the phase
3. Precision of the phase measurement
measurement
coordinate measurements
From a purely formal point of view, the phase of a harmonic oscillator sp (as well as any other physical observable) can be measured with any necessary precision. A specific feature of the phase is that its canonically conjugate observable - number of quanta N in the oscillator - is a non-negative value. This will by no means affect the process of measurement until the precision of measument is not too high,
and therefore the perturbation of the number of quanta is sufficiently small,
( Niinit is the mean number of quanta before the measurement) . Further improvement of the measurement precision would lead not only to the increase of the uncertainty of the number of quanta, but also to the increase of its mean value to provide the fulfillment of the inequality ANPen < N.
The existing methods of phase measurement are
based on the following simple principle. Let the oscillations of the oscillator coordinate be represented as X(t) = (A+Xl)cos(wt)
+X~sin(wt),
where A is a given large value (classical amplitude of the oscillations),
where the X1,2 are the quadrature amplitudes. Then the measurement of the phase can be reduced to the measurement of the quadrature amplitude X2, because in this case
(see also the Appendix). It is easy to show that the measurement of X2 at the level of the standard quantum limit (SQL),
AX2 = AXsoL = d= (m is the oscillator mass, w is its eigenfrequency) corresponds to the phase measurement with the uncertainty 1
The required additional energy must apparently be provided by the meter. It is evident therefore that the higher the precision of the measurement, the larger the mean number of quanta after the measurement, Nina,,
(1) where 1st1 is the first zero of the Airy function. Formula ( 1) should be considered as a fundamental limit for the precision of the measurement of the phase in a harmonic oscillator.
(2:
A4os.j~= 2TP
(i? is the mean number of quanta). By reducing AXz, one can improve the precision of the phase measurement, but only up to a level providing the fulfillment of the relation
(Axl),n = A more precise relation, valid for sufficiently large R (N 2 50 ) , is given by the following formula [ 41,
by
n amwAX
( (AX1 )pert is the perturbation of the quadrature amplitude Xt under measurement). The ultimate precision of the measurement of the phase (as it is shown in Ref. [ 51) is Arp = Jln R/2N
(3)
(this formula is valid if A >> 1). Comparing formulas ( 1) and (3), we can conclude that the coordinate measurements allow one to achieve
V.B. Bragin.+
a precision in the measuremnt of the phase that differs from the principal limit ( 1) only by a numerical factor of the order of unity. This circumstance makes the measurement of the phase substantially different, for example, from the measurement of the oscillator energy. Simultaneously, the method of phase measurement considered has a substantial disadvantage: the higher the required precision of the measurement, the narrower the dynamic range of the possible phase values to be measured. To demonstrate this, let us consider a simple example. Let the oscillator be prepared in a squeezed state with respect to the quadrature amplitude X2. Let an external action shift the phase of its oscillations by the value (p << 1. The experimentalist tries to determine sp by measuring X2. It can be easily shown that the dispersion of X;! in the new state is equal to W2kw
=
(AXz)2cos2~+
(Ax2)nw= J (AX2j2 A
-AX,,,
< X < AXmeaS, AX,,,
+ (A&J24p2
A
(4)
The value A9 substantially depends on 9, the stronger the higher the squeezing of the initial state. If the experimentalist has a priori information on the maximal value of 9, then this information can be used for choosing the optimal squeezing coefficient,
GCA&QL,
and “no” outside this interval. It is evident that during a time equal to half-period of the oscillations after turning on such a “null-detector”, it will give a single short pulse with duration r=- 2AX,,, WA (A is the amplitude of the oscillations, w the frequency ) , when the oscillator coordinate crosses zero.
This allows one to determine the phase of the oscillations with the uncertainty Apmeas =
where AX], AX2 are the initial uncertainties. Therefore, the measurement error for (p will be equal to =
of dynamic range limitations. Let the experimentalist have a sensor of the oscillator coordinate with a strongly nonlinear response reading “yes” if the oscillator coordinate is inside the interval
(AXi)2sin2cp
N (AX2)2 + (AX, )2rp2,
Ap
3
et al. /Physics Letters A 202 (1995) l-6
AXme,
+T = -
A
5‘
2fi’
bwas
AXneas
@SQL
AXSQL ’
5 =-=-
=-
During the interaction, the momentum of the oscillator will be perturbed by APFn = F.
/i meas
The resulting perturbation of the oscillator energy is ii&A
AEpen = wAAPpen= 2AX,,,
AX2
(6)
in full accordance with the uncertainty relation In the worst case lspj= vo,, we obtain Aip=J
2AXzAXi ~Pmax A
=&zz
i.e. a measurement precision close to the limit (3) is achieved only within a very narrow interval of phase values, (omax6 l/i?.
AE,,, = +o. A‘PmeZis
(7)
It should be mentioned that the semi-classical analysis above is valid only for oscillator states with a very small overlap with the ground state. In particular, the following condition must be fulfilled, -Ap,rr < A .
4. The null-detector Below we consider the scheme of a phase meter that allows one to reach the precision limit (3) and is free
mw
It is possible to show that this relation imposes the former limitation (3) on the precision of the phase measurement.
4
V.B. Braginsky et al, /Physics
Attempts of a more precise measurement would lead to a nontrivial dynamic behavior of the oscillator: occurrence of a nonzero probability for the oscillator coordinate that does not cross zero at all [ 61. (The smaller AX,,, , the larger this probability.) The oscillator will be “reflected’ from the plane X = 0 and start moving in opposite direction. This phenomenon is essentially a form of the quantum Zeno effect [ 7,8]. As already mentioned above, the phase of the harmonic motion is a QND variable. Therefore the measurement procedure described above can be repeated more than once, for example, for the detection of a small phase-shifting external action on the oscillator. For a large number of repetitions, however, another dynamical effect - heating of the oscillator - must be observed. This is due to the fact that in addition to the random perturbation of the oscillator energy (6)) during each measurement its mean energy is increased by
Letters A 202 (I 995) I-6
the necessary perturbation of the momentum of the mechanical oscillator. The precision of the phase measurement in this scheme is equal to
where AXtranspis the coordinate interval, inside which the Fabry-Perot resonator is transparent,
A is the mean number of photons registered by the photodetector, R is the reflectivitty of the mirror, A is the wavelength. Let us estimate the feasibility of experimental realization of this scheme. Let the oscillator be a cube of mass m N 10e9 g suspended in a trap providing the frequency of the mechanical oscillations w N lo3 S -I. For such an oscillator
AXSQL= 5. Examples
of realization of phase meters
J
&
N 2 x lo-”
cm.
On the other hand, within this scheme, 5.1. Mechanical system AXW, = lo-” For mechanical systems, measurements at a level of resolution close to the standard quantum limit (or better) are a complicated experimental problem because of the extremely small number of mechanical quanta. However, it is possible to propose a not entirely gedanken variant for the realization of a null-detector. Let the mass m of a mechanical oscillator be equipped with a rigidly bound mirror, and let this mirror form a Fabry-Perot resonator together with another immobile mirror. Let the pump light be fed into the Fabry-Perot from the side of the immobile mirror, and a photodetector be placed behind the moving mirror. Let us assume, in addition, that the frequency of light is chosen in such a way that the interferometer is tuned to resonance when the oscillator coordinate crosses zero. Then the interferometer will transmit’ light to the photodetector only when X is close to zero. On the other hand, only at these instants the interferometer will contain a substantial amount of light energy. Fluctuations of this stored energy (caused by the photon shot noise) will provide
cm
6x l~-scm)(5t(;~7)’
Therefore, to provide AX,,, < AXSQLand thereby to break through the SQL in phase measurement, it is sufficient to have 1 - R < 5 x 10m7- only two times better than in existing mirrors [ 91. It should be mentioned that this concrete realization of the null-detector possesses an additional inherent error of phase measurement, associated with the random delay of the registered photons due to finite relaxation time of the optical resonator, 7* =
L
c(1 - R)’ where L is the distance between the mirrors. The value of this additional error (purely technical in origin) is equal to
Note that A@d,j < ApPmeas if the amplitude of the mechanical oscillations is not very large. (For the
5
V.B. Braginsky et al. /Physics Letters A 202 (1995) I-4
above numerical example A < lo-’ cm, which corresponds to the mean number of mechanical quanta iVm< 10’2.) 5.2.
Microwave
system
At present, well-elaborated electron beam techniques in combination with high achieved values of the quality factor in superconducting microwave cavities allow one to propose a realistic scheme for the QND measurement of the quadrature amplitude in a microwave resonator. Let a klystron-type resonator with well-localized capacity C be probed by a short cluster of electrons passing through the capacity gap across the electrical field lines. The length of this cluster should fulfill the condition Z/v << 7” where v is the velocity of the electrons, T,= 2r/oe is the period of the resonator oscillations. The transit time necessary to cross the capacity gap must not exceed ;T,. Then the additional transverse momentum acquired by the electrons during the transit will be proportional to one of the two quadrature amplitudes of the e.m. field in the resonator XI. By measuring the deflection angle of the electrons (Y, the experimentalist will be able to determine the value of this quadrature amplitude. On the other hand, scatter in the values of the y-coordinates of the electrons will lead to the perturbation of the other quadrature amplitude X2. It is easy to show that the product of the measurement error AXt measand the perturbation AX, rert will correspond to the uncertainty relation
quadrature amplitude. (The maximal number of repetitions is limited only by the value of the resonator quality factor.) The quadrature amplitude measurement scheme described above can be the basis of a null-detector for a microwave oscillator. To this end, first, the period between “launched” clusters of probe electrons should be Te+ A4p/we, where Aqpis the necessary precision of the phase measurement. Second, the output detector should “read’ only non-deflected electrons (i.e. those passing through the resonator at the instant of zero field). Third, all other electrons should be sent through the resonator again with the phase shift of r. Then the information on the field state, stored in the ycomponent of the electron momentum, will be erased. The perturbation of the resonator field by the electron during the first passage will be simultaneously compensated.
Appendix A
Here we will consider the connection between the coordinate representation and phase representation provided AN >> 1, PAX//i >> 1, X = 0 for the state of the oscillator and will show that in this case the phase measurement coincides with the coordinate measurement. Let us consider the state which in the coordinate representation has the following form, $(X) = (xl@) = a
AXI mzJX2
(A.1)
pert = $
(p is the impedance of the resonator) when the uncertainties of the y-coordinates and momenta of the electrons correspond to the diffraction limit,
where a(x), S(X) ace real functions, X = 0. In the number basis the state (A.l) is determined as
Ayhp, = $Fi. Simple estimates have shown that for realistic parameters of the resonator (capacity gap 21 lo-* cm, capacity value N 0.1 pF), it is possible to detect the deflection of a single electron caused by zero-point oscillations in the resonator. By repeating the measurement several times with the period Te,one can improve the precision of the measurement and obtain an even stronger squeezed state with respect to a given
where (n[x) =
(4
l
-!-
AXc)tj4 mH”
Here the H,,(x) are Hermite polynomials and
6
V.B. Braginsky et al. /Physics Letters A 202 (1995) I-6
AX, = d?$%&
APC = JG.
e/(e) =
With the asymptotics of the Her-mite polynomials and the factorial for large n, H,(x)
p&(Bg-)
xexp [i( *,(0/z)
-8’&)].
N Jz (2n/e)“/*
x exp(ix*)
cos(&Gx
The plus sign corresponds to P > 0, the minus to P>O,and
- +n),
n! 2 dG(n/e)n,
8’ = -(e
=e++,
we obtain for n >> 1 114
References
M J
a(t) exp[ iS(AXt) - iPAXr/n]
x exp( iyt) dt. The state (A.l) in the phase representation [2] is co
ccl(@=
if P < 0.
(A.2)
where Z(Y) = &__
if P > 0,
We can easily see that the wavefunction of the oscillator has the same form in phase and coordinate representations. Therefore, the measurement of the coordinate for the case AN > 1, PAX/n > 1, X = 0 coincides with the phase measurement.
X
+exp($-)c[(&-fi)z]},
- +),
-$==~bev(-in@.
Since AN >> 1, PAX/A >> 1, one can replace the sum by the integral and use the approximate formula (A.2) for &, . After performing the inverse Fourier transform. we obtain the solution
[l] P Carruthers and M.M. Nieto, Rev. Mod. Phys. 40 (1968) 411. [2] WI? Schleich and SM. Barnett, eds., Quantum phase and phase dependent measurements, Phys. Ser. T48 ( 1993). [3] V.B. Braginsky and P.Ya. Khalili, Quantum measurement, ed. KS. Thome (Cambridge Univ. Press, Cambridge, 1992). [4] A.A. Kulaga and EYa. Khalili, Sov. Phys. JETP 104 ( 1993) 3358. [5] M.J. Collett, Phys. Ser. T48 (1993) 124. [ 61 A.A. Kulaga, Phys. Lett. A 202 (1995) 7. [7] L.A. Khaltin, Sov. Phys. Usp. 33 (1990) 10. [8] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18 (1977) 758. [ 91 G. Rempe, R.J. Thompson and H.J. Kimble, Opt. Lett. 17 (1992) 363.
EISEVIER
A
Physics Letters A 202 (1995) l-6
Quantum-non-demolition measurement of the phase V.B
. Braginsky ‘, EYa. Khalili, A.A. Kulaga
Chair of Molecular Physics and Physical Measurements, Department of Physics, Moscow State University, 119899 Moscow, Russian Federation
Received 4 March 1995; accepted for publication 17 March 1995 Communicated by V.M. Agranovich
Abstract The problem of the QND measurement of the phase of a linear quantum oscillator is analyzed. It is shown that by measuring the quadrature amplitude one can reach a resolution close to the fundamental limit in a narrow range of phase values. A new method of QND phase measurement is proposed and analyzed. This method is also based on the coordinate interaction between the quantum object and the measuring device but is free from the limitation mentioned above. Several practical schemes of phase measurementsare discussed.
1. Introduction The group of problems associated with the phase observable has been for a long time, and partially remained until now, an incomplete part of quantum theory. The main reason for this is the absence of a “good” operator of the phase defined on the entire space of oscillator states. The analysis of the mathematical aspects of this problem can be found in Ref. [ 11. In the last 20 years, the formal quantum theory of the phase has been substantially developed (see Ref. [ 21). However, presently a complete quantummechanical description of the phase as a physical quantity does not exist. Such a description must necessarily include a definition (at least gedanken) of its measurement procedure. In this area, the progress has not been impressive. Till now, to the authors’ knowledge, not a single (even gedanken) procedure of “true phase” measurement has been proposed, which under
1E-mail: [email protected].
ideal conditions would reduce the wavefunction of the oscillator to a state with a given phase. The realization of such a scheme would allow one, for example, to substantially improve one of the basic methods of spectroscopy - collation of frequencies of two oscillators (one of them can be the basic element in a high-stability self-oscillator). It can be also mentioned that the phase of the harmonic oscillator is a QND observable [ 31 and therefore allows, in principle, for a continuous non-perturbative monitoring. Apparently, it is the single simple example of the QND observables that is not an integral of motion. The difficulty of realization of a “true phase” measurement is that the meter should respond only to the phase of the oscillations and should in no way respond to the amplitude. One can hardly imagine a physical interaction that would meet the requirement above. In this paper, we consider two methods for measuring the oscillator phase. Although based on the usual (coordinate) interaction of the meter with the object under study, they allow one to obtain a precision of the phase measurement close to the theoretical limit.
0375-9601/95/@9.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO375-9601(95)00279-O
2
V.B. Braginsky et al. /Physics
Letters A 202 (I 995) I-6
2. On the ultimate precision in the phase
3. Precision of the phase measurement
measurement
coordinate measurements
From a purely formal point of view, the phase of a harmonic oscillator sp (as well as any other physical observable) can be measured with any necessary precision. A specific feature of the phase is that its canonically conjugate observable - number of quanta N in the oscillator - is a non-negative value. This will by no means affect the process of measurement until the precision of measument is not too high,
and therefore the perturbation of the number of quanta is sufficiently small,
( Niinit is the mean number of quanta before the measurement) . Further improvement of the measurement precision would lead not only to the increase of the uncertainty of the number of quanta, but also to the increase of its mean value to provide the fulfillment of the inequality ANPen < N.
The existing methods of phase measurement are
based on the following simple principle. Let the oscillations of the oscillator coordinate be represented as X(t) = (A+Xl)cos(wt)
+X~sin(wt),
where A is a given large value (classical amplitude of the oscillations),
where the X1,2 are the quadrature amplitudes. Then the measurement of the phase can be reduced to the measurement of the quadrature amplitude X2, because in this case
(see also the Appendix). It is easy to show that the measurement of X2 at the level of the standard quantum limit (SQL),
AX2 = AXsoL = d= (m is the oscillator mass, w is its eigenfrequency) corresponds to the phase measurement with the uncertainty 1
The required additional energy must apparently be provided by the meter. It is evident therefore that the higher the precision of the measurement, the larger the mean number of quanta after the measurement, Nina,,
(1) where 1st1 is the first zero of the Airy function. Formula ( 1) should be considered as a fundamental limit for the precision of the measurement of the phase in a harmonic oscillator.
(2:
A4os.j~= 2TP
(i? is the mean number of quanta). By reducing AXz, one can improve the precision of the phase measurement, but only up to a level providing the fulfillment of the relation
(Axl),n = A more precise relation, valid for sufficiently large R (N 2 50 ) , is given by the following formula [ 41,
by
n amwAX
( (AX1 )pert is the perturbation of the quadrature amplitude Xt under measurement). The ultimate precision of the measurement of the phase (as it is shown in Ref. [ 51) is Arp = Jln R/2N
(3)
(this formula is valid if A >> 1). Comparing formulas ( 1) and (3), we can conclude that the coordinate measurements allow one to achieve
V.B. Bragin.+
a precision in the measuremnt of the phase that differs from the principal limit ( 1) only by a numerical factor of the order of unity. This circumstance makes the measurement of the phase substantially different, for example, from the measurement of the oscillator energy. Simultaneously, the method of phase measurement considered has a substantial disadvantage: the higher the required precision of the measurement, the narrower the dynamic range of the possible phase values to be measured. To demonstrate this, let us consider a simple example. Let the oscillator be prepared in a squeezed state with respect to the quadrature amplitude X2. Let an external action shift the phase of its oscillations by the value (p << 1. The experimentalist tries to determine sp by measuring X2. It can be easily shown that the dispersion of X;! in the new state is equal to W2kw
=
(AXz)2cos2~+
(Ax2)nw= J (AX2j2 A
-AX,,,
< X < AXmeaS, AX,,,
+ (A&J24p2
A
(4)
The value A9 substantially depends on 9, the stronger the higher the squeezing of the initial state. If the experimentalist has a priori information on the maximal value of 9, then this information can be used for choosing the optimal squeezing coefficient,
GCA&QL,
and “no” outside this interval. It is evident that during a time equal to half-period of the oscillations after turning on such a “null-detector”, it will give a single short pulse with duration r=- 2AX,,, WA (A is the amplitude of the oscillations, w the frequency ) , when the oscillator coordinate crosses zero.
This allows one to determine the phase of the oscillations with the uncertainty Apmeas =
where AX], AX2 are the initial uncertainties. Therefore, the measurement error for (p will be equal to =
of dynamic range limitations. Let the experimentalist have a sensor of the oscillator coordinate with a strongly nonlinear response reading “yes” if the oscillator coordinate is inside the interval
(AXi)2sin2cp
N (AX2)2 + (AX, )2rp2,
Ap
3
et al. /Physics Letters A 202 (1995) l-6
AXme,
+T = -
A
5‘
2fi’
bwas
AXneas
@SQL
AXSQL ’
5 =-=-
=-
During the interaction, the momentum of the oscillator will be perturbed by APFn = F.
/i meas
The resulting perturbation of the oscillator energy is ii&A
AEpen = wAAPpen= 2AX,,,
AX2
(6)
in full accordance with the uncertainty relation In the worst case lspj= vo,, we obtain Aip=J
2AXzAXi ~Pmax A
=&zz
i.e. a measurement precision close to the limit (3) is achieved only within a very narrow interval of phase values, (omax6 l/i?.
AE,,, = +o. A‘PmeZis
(7)
It should be mentioned that the semi-classical analysis above is valid only for oscillator states with a very small overlap with the ground state. In particular, the following condition must be fulfilled, -Ap,rr < A .
4. The null-detector Below we consider the scheme of a phase meter that allows one to reach the precision limit (3) and is free
mw
It is possible to show that this relation imposes the former limitation (3) on the precision of the phase measurement.
4
V.B. Braginsky et al, /Physics
Attempts of a more precise measurement would lead to a nontrivial dynamic behavior of the oscillator: occurrence of a nonzero probability for the oscillator coordinate that does not cross zero at all [ 61. (The smaller AX,,, , the larger this probability.) The oscillator will be “reflected’ from the plane X = 0 and start moving in opposite direction. This phenomenon is essentially a form of the quantum Zeno effect [ 7,8]. As already mentioned above, the phase of the harmonic motion is a QND variable. Therefore the measurement procedure described above can be repeated more than once, for example, for the detection of a small phase-shifting external action on the oscillator. For a large number of repetitions, however, another dynamical effect - heating of the oscillator - must be observed. This is due to the fact that in addition to the random perturbation of the oscillator energy (6)) during each measurement its mean energy is increased by
Letters A 202 (I 995) I-6
the necessary perturbation of the momentum of the mechanical oscillator. The precision of the phase measurement in this scheme is equal to
where AXtranspis the coordinate interval, inside which the Fabry-Perot resonator is transparent,
A is the mean number of photons registered by the photodetector, R is the reflectivitty of the mirror, A is the wavelength. Let us estimate the feasibility of experimental realization of this scheme. Let the oscillator be a cube of mass m N 10e9 g suspended in a trap providing the frequency of the mechanical oscillations w N lo3 S -I. For such an oscillator
AXSQL= 5. Examples
of realization of phase meters
J
&
N 2 x lo-”
cm.
On the other hand, within this scheme, 5.1. Mechanical system AXW, = lo-” For mechanical systems, measurements at a level of resolution close to the standard quantum limit (or better) are a complicated experimental problem because of the extremely small number of mechanical quanta. However, it is possible to propose a not entirely gedanken variant for the realization of a null-detector. Let the mass m of a mechanical oscillator be equipped with a rigidly bound mirror, and let this mirror form a Fabry-Perot resonator together with another immobile mirror. Let the pump light be fed into the Fabry-Perot from the side of the immobile mirror, and a photodetector be placed behind the moving mirror. Let us assume, in addition, that the frequency of light is chosen in such a way that the interferometer is tuned to resonance when the oscillator coordinate crosses zero. Then the interferometer will transmit’ light to the photodetector only when X is close to zero. On the other hand, only at these instants the interferometer will contain a substantial amount of light energy. Fluctuations of this stored energy (caused by the photon shot noise) will provide
cm
6x l~-scm)(5t(;~7)’
Therefore, to provide AX,,, < AXSQLand thereby to break through the SQL in phase measurement, it is sufficient to have 1 - R < 5 x 10m7- only two times better than in existing mirrors [ 91. It should be mentioned that this concrete realization of the null-detector possesses an additional inherent error of phase measurement, associated with the random delay of the registered photons due to finite relaxation time of the optical resonator, 7* =
L
c(1 - R)’ where L is the distance between the mirrors. The value of this additional error (purely technical in origin) is equal to
Note that A@d,j < ApPmeas if the amplitude of the mechanical oscillations is not very large. (For the
5
V.B. Braginsky et al. /Physics Letters A 202 (1995) I-4
above numerical example A < lo-’ cm, which corresponds to the mean number of mechanical quanta iVm< 10’2.) 5.2.
Microwave
system
At present, well-elaborated electron beam techniques in combination with high achieved values of the quality factor in superconducting microwave cavities allow one to propose a realistic scheme for the QND measurement of the quadrature amplitude in a microwave resonator. Let a klystron-type resonator with well-localized capacity C be probed by a short cluster of electrons passing through the capacity gap across the electrical field lines. The length of this cluster should fulfill the condition Z/v << 7” where v is the velocity of the electrons, T,= 2r/oe is the period of the resonator oscillations. The transit time necessary to cross the capacity gap must not exceed ;T,. Then the additional transverse momentum acquired by the electrons during the transit will be proportional to one of the two quadrature amplitudes of the e.m. field in the resonator XI. By measuring the deflection angle of the electrons (Y, the experimentalist will be able to determine the value of this quadrature amplitude. On the other hand, scatter in the values of the y-coordinates of the electrons will lead to the perturbation of the other quadrature amplitude X2. It is easy to show that the product of the measurement error AXt measand the perturbation AX, rert will correspond to the uncertainty relation
quadrature amplitude. (The maximal number of repetitions is limited only by the value of the resonator quality factor.) The quadrature amplitude measurement scheme described above can be the basis of a null-detector for a microwave oscillator. To this end, first, the period between “launched” clusters of probe electrons should be Te+ A4p/we, where Aqpis the necessary precision of the phase measurement. Second, the output detector should “read’ only non-deflected electrons (i.e. those passing through the resonator at the instant of zero field). Third, all other electrons should be sent through the resonator again with the phase shift of r. Then the information on the field state, stored in the ycomponent of the electron momentum, will be erased. The perturbation of the resonator field by the electron during the first passage will be simultaneously compensated.
Appendix A
Here we will consider the connection between the coordinate representation and phase representation provided AN >> 1, PAX//i >> 1, X = 0 for the state of the oscillator and will show that in this case the phase measurement coincides with the coordinate measurement. Let us consider the state which in the coordinate representation has the following form, $(X) = (xl@) = a
AXI mzJX2
(A.1)
pert = $
(p is the impedance of the resonator) when the uncertainties of the y-coordinates and momenta of the electrons correspond to the diffraction limit,
where a(x), S(X) ace real functions, X = 0. In the number basis the state (A.l) is determined as
Ayhp, = $Fi. Simple estimates have shown that for realistic parameters of the resonator (capacity gap 21 lo-* cm, capacity value N 0.1 pF), it is possible to detect the deflection of a single electron caused by zero-point oscillations in the resonator. By repeating the measurement several times with the period Te,one can improve the precision of the measurement and obtain an even stronger squeezed state with respect to a given
where (n[x) =
(4
l
-!-
AXc)tj4 mH”
Here the H,,(x) are Hermite polynomials and
6
V.B. Braginsky et al. /Physics Letters A 202 (1995) I-6
AX, = d?$%&
APC = JG.
e/(e) =
With the asymptotics of the Her-mite polynomials and the factorial for large n, H,(x)
p&(Bg-)
xexp [i( *,(0/z)
-8’&)].
N Jz (2n/e)“/*
x exp(ix*)
cos(&Gx
The plus sign corresponds to P > 0, the minus to P>O,and
- +n),
n! 2 dG(n/e)n,
8’ = -(e
=e++,
we obtain for n >> 1 114
References
M J
a(t) exp[ iS(AXt) - iPAXr/n]
x exp( iyt) dt. The state (A.l) in the phase representation [2] is co
ccl(@=
if P < 0.
(A.2)
where Z(Y) = &__
if P > 0,
We can easily see that the wavefunction of the oscillator has the same form in phase and coordinate representations. Therefore, the measurement of the coordinate for the case AN > 1, PAX/n > 1, X = 0 coincides with the phase measurement.
X
+exp($-)c[(&-fi)z]},
- +),
-$==~bev(-in@.
Since AN >> 1, PAX/A >> 1, one can replace the sum by the integral and use the approximate formula (A.2) for &, . After performing the inverse Fourier transform. we obtain the solution
[l] P Carruthers and M.M. Nieto, Rev. Mod. Phys. 40 (1968) 411. [2] WI? Schleich and SM. Barnett, eds., Quantum phase and phase dependent measurements, Phys. Ser. T48 ( 1993). [3] V.B. Braginsky and P.Ya. Khalili, Quantum measurement, ed. KS. Thome (Cambridge Univ. Press, Cambridge, 1992). [4] A.A. Kulaga and EYa. Khalili, Sov. Phys. JETP 104 ( 1993) 3358. [5] M.J. Collett, Phys. Ser. T48 (1993) 124. [ 61 A.A. Kulaga, Phys. Lett. A 202 (1995) 7. [7] L.A. Khaltin, Sov. Phys. Usp. 33 (1990) 10. [8] B. Misra and E.C.G. Sudarshan, J. Math. Phys. 18 (1977) 758. [ 91 G. Rempe, R.J. Thompson and H.J. Kimble, Opt. Lett. 17 (1992) 363.