An improvement in precision beyond the limit set by the vacuum-state or zero-point fluctuations of the electromagnetic field is reported for the detection of amplitude modulation encoded on a weak signal beam. The improvement is achieved by employing the squeezed light from an optical parametric oscillator to reduce the level of fluctuations below the shot-noise limit. An increase in signal-to-noise ratio of 2.5 dB relative to the shot-noise limit is demonstrated

Kimble, H. J.

Wu, Ling-An

Xiao, Min

English

Caltech Authors - Main

476 OPTICS LETTERS / Vol. 13, No. 6 / June 1988
Detection of amplitude modulation with squeezed light for
sensitivity beyond the shot-noise limit
Min Xiao, Ling-An Wu, and H. J. Kimble
Department of Physics, University of Texas at Austin, Austin, Texas 78712
Received January 22, 1988; accepted March 21, 1988
An improvement in precision beyond the limit set by the vacuum-state or zero-point fluctuations of the electromag-
netic field is reported for the detection of amplitude modulation encoded on a weak signal beam. The improvement
is achieved by employing the squeezed light from an optical parametric oscillator to reduce the level of fluctuations
below the shot-noise limit. An increase in signal-to-noise ratio of 2.5 dB relative to the shot-noise limit is
demonstrated.
An exploration of the limits on the detection of small
changes in amplitude of the electromagnetic field is of
fundamental and practical importance in optical
physics. While a variety of noise sources of technical
origin often limit sensitivity, the fundamental limit on
the detection of a small absorption or loss y has been
the so-called shot-noise limit (SNL), which sets a low-
er bound on the minimum detectable change 5A of the
coherent amplitude A of the field given by -y _ 6A/A
1//N, where N is the number of photons recorded in
the time interval of the measurement. A diverse set of
experiments in ultrasensitive laser spectroscopy1 pro-
vides but one example in which it has been possible to
overcome technical noise, to eliminate extraneous co-
herent signals, and to reach the SNL. However, with
conventional laser light in a coherent state, it is not
possible to proceed further to a sensitivity greater
than that specified by the SNL.
Improvements in precision beyond the SNL can be
realized by employing squeezed states of light. A
squeezed state is characterized by a phase-dependent
redistribution of quantum fluctuations such that the
variance of one of two quadrature components of the
field is reduced below that of the vacuum state. By
encoding the signal (e.g., amplitude modulation) on
the field variable with reduced fluctuations and by
employing a detection scheme that is largely insensi-
tive to the increased fluctuations of the conjugate field
variable, sensitivity greater than the SNL can be
achieved.3' 4
In this Letter we report experiments that utilize
squeezed light to improve the sensitivity for the detec-
tion of amplitude modulation by 2.5 dB relative to the
shot-noise or vacuum-state limit. In the measure-
ments a weak signal beam is combined with either a
vacuum-state or a squeezed-state input; the amplitude
of the resulting beam is modulated in propagation
through an acousto-optic modulator. The observed
increase in sensitivity for the detection of the signal
modulation is currently limited principally by simple
linear losses in propagation and detection and not by
the degree of squeezing available from our source.5' 6
Our apparatus is depicted in general outline in Fig.
1. Consider first the case of the inputs to m1 being a
coherent state -o and a vacuum state P,, with the
transmissivity of m1 being small.
7 The weak probe
beam t1 that is output from m1 is modulated in ampli-
tude at frequency go by an acousto-optic modulator
(AOM) to generate a field t 2 given by8
(A) = (E1 ) T COs 0(t) (1)
where 0(t) = So + 01 COS Qot and is proportional to the
amplitude of the index of refraction variation pro-
duced by the traveling acoustic wave in the modulator.
T is the static transmission coefficient (0 = 0) of the
AOM. For the case of low-loss operation of the modu-
lator we expand Eq. (1), keeping only lowest-order
terms at the modulation frequency Qo to find that
(E 2 ) _ (E 1 ) T(1 - 6 cos Q00, (2)
where the modulation index 6 = 01 sin 00. The modu-
lated field 22 enters the signal port of a balanced
homodyne detector formed by the beam splitter m2
OPO A
, , Es
A
ELO'
E0
Di
12
Fig. 1. Schematic of the apparatus for detection of ampli-
tude modulation with squeezed states. The reflectivity of
ml is close to unity; m 2 is a beam splitter with R = T = 0.5.
0146-9592/88/060476-03$2.00/0 © 1988, Optical Society of America
n -- |I AOM
June 1988 / Vol. 13, No. 6 / OPTICS LETTERS 477
(50-50) and the photodiodes D1 and D2 and operated
such thatI(PLO)I» I(A12)I.9 The amplified photocur-
rents il and i2 from the two arms are combined with a
1800 phase shift to produce the photocurrent i, which
is directed to a commercial spectrum analyzer.
The photocurrent i contains a signal component is at
frequency 9o of rms amplitude
i,(Qo) = 2Tea- (-PLO) (EA) 6 COS3, (3)
where the units of the fields are chosen such that
(P.t) expresses a flux in photons per second across
the detector area. The quantum efficiency of the pho-
todiodes is given by ar, while X expresses the hetero-
dyne efficiency for overlap of the fields E2 and tLO- 3
is the phase between (E2 ) and (BLO). For d = +=
pwr (p an integer), the fields are in phase, and the
amplitude modulation encoded on E2 results in ampli-
tude modulation of ELO. For d = 0_ = (2p + 1)r/2,
the modulation of P2 results in phase modulation of
BLO and hence produces no signal current to lowest
order in 6. In addition to the signal current i,(Q0),
there is a noise current in against which any coherent
detection must be made. In the case of a vacuum-
state input for ES, in is just the shot-noise current of
rms amplitude
in = (2eiB)1 12 = (2e2 a (ELOttLO) B)1 2 . (4)
T'(Q0o, ) = TI/[1 + AS(Qo, ct)] = cP26
2 cos 2 0 (6)
where r represents the overall efficiency for propaga-
tion and detection of the squeezed light.6 The two
phases (S, O) appearing in Eq. (6) specify, respectively,
the orientation of the coherent modulation at jo and
the orientation of the squeezed-noise ellipse relative to
the local-oscillator field. For 0 = 0+ = q7r (q an inte-
ger), the enhanced fluctuations S+ of the squeezed
state align with (PLO), while for 0 = 0- = (2q + 1)ir/2
the reduced fluctuations S_ of the squeezed state align
with (ELO). From Eq. (6) we see that for 0 = 0-, an
improvement in signal-to-noise ratio beyond the shot-
noise or vacuum-state limit is given by the factor (1 +
PS_)-1>> 1 for S& -1 and v 1. On the other hand,
for 0 = 0+, there is a large degradation in the signal-to-
noise ratio by the factor (1 + PS+)-1 << 1, pointing out
the requirement for precise control of 0.
D(dB)
0The signal-to-noise ratio for coherent-state inputs (P0,
PLO) and a vacuum-state input P, is thus
TV = is2 aP 2 cos 
2
2i2 B (5)
where we have defined the effective power P2
Tnq2I(_ 1)j2 (photons/sec). For ,3 = 3+, a signal-to-
noise ratio of unity (TV = 1) implies a minimum de-
tectable modulation index 6 = (B/aP 2)1 /2 = 11VN2,
with N2 as the mean number of photoelectrons in the
measurement interval B-1 due to P2. This limit on the
minimum detectable amplitude modulation is the
SNL discussed above with the association of 6 with y.
Any improvement beyond this limit is not possible in
conventional detection schemes with coherent states.
An improvement in sensitivity is possible, however,
if a squeezed field ES is injected in place of the vacuum
field at m1 to reduce the noise against which the modu-
lation signal must be detected. The squeezed light in
our experiment is generated by a subthreshold optical
parametric oscillator with degenerate signal and idler
waves at X = 1.064 Am. 5 ,6 The output field PL from the
optical parametric oscillator is characterized by a
spectrum of squeezing S(Q, q), where Q is the frequen-
cy offset from the carrier and p selects the quadrature
phase. 10 S(Q2, k) is normalized such that S = 0 for a
vacuum state, while for a perfectly squeezed field, S -
-1 for one quadrature and S -- for the conjugate
quadrature. With the injection of squeezed vacuum
at m1 and under the assumptions of modest squeezing
and small modulation index, the calculation of the
signal current proceeds exactly as before [Eq. (3)].
However, the noise in the difference photocurrent now
becomes in2(Q, ck) = in 2[1 + AS(Q, 1)], resulting in a
signal-to-noise ratio
D (d B)
A= 7r Phase :
Fig. 2. Spectral density 4? of photocurrent fluctuations ver-
sus signal phase 3 for fixed analysis frequency Qo/27r = 1.6
MHz. The phase f between (E2) and (ELO) is swept in a
sawtooth fashion to produce the periodic variation in 4? with
the sharp features corresponding to the flyback to restart
the scan. The dashed line gives the vacuum level obtained
with inputs E0 and ES blocked. The gap in (a) is caused by a
trigger event that interrupts the local-oscillator beam. (a)
Vacuum-state input for E5, (b) squeezed-state input for E5.
The slow variation of (D over the scan in (b) results from a
variation in 0 to adjust for minimum noise level. An im-
provement of 2.5 dB in signal-to-noise ratio is achieved in (b)
relative to (a) near the noise minimum. The time for the
entire trace is 0.2 sec.
478 OPTICS LETTERS / Vol. 13, No. 6 / June 1988
Our experimental results for the improvement in
sensitivity for the detection of amplitude modulation
are displayed in Fig. 2, which records the spectral
density 4? of photocurrent fluctuations of the differ-
ence photocurrent i as a function of phase /. The
periodic up-down variation of 4? with / reflects the cos /
dependence of the signal photocurrent as given by Eq.
(3). For : = A+, the signal modulation is detected with
maximum efficiency (cos2 d = 1), and the detected
level 4' is the sum of contributions from the coherent
modulation and the noise current. For / = A-, the
modulation is predominantly frequency modulation
with respect to the local-oscillator field, and no coher-
ent signal is expected (cos2 / = 0). Hence the detected
level in this situation should correspond to photocur-
rent noise alone, as given by 4? = 10 log (1 + CS). In
Fig. 2(a) the input A, to ml is a vacuum state for which
S = 0, while in Fig. 2(b) a squeezed state is injected for
which S < 0 for 1 properly chosen. The actual vacu-
um-state or shot-noise level in Fig. 2 is indicated by a
dashed line and is obtained with the inputs A, and Po
blocked after the procedure discussed in Ref. 6. Fig-
ure 2(a) corresponds roughly to the case T'I 1 (signal-
to-noise ratio of unity) and represents the SNL for
signal detection. A decrease in noise level and an
increase in signal-to-noise ratio beyond this limit are
shown in Fig. 2(b). Figures 2(a) and 2(b) are recorded
under identical conditions with the single exception
that squeezed-vacuum light is injected into ml for Fig.
2(b). In Fig. 2(b) the angle 0 is slowly varied to reach
a position of minimum noise. On the left-hand edge
of the figure the noise level is well in excess of the SNL.
As a value near 1 = 0. is reached in the center of the
figure, there is a decrease in photocurrent noise below
the SNL. From a number of traces as in Fig. 2, we find
an improvement in signal-to-noise ratio relative to the
vacuum-state limit of (2.5 t 0.5) dB (a factor of 1.8 in
terms of noise power). The consequence of this im-
provement is that a given level of precision for i,(Qo)
could be reached in a measurement time shortened by
a factor <1.8 relative to shot-noise-limited measure-
ments. Alternatively, for a fixed measurement inter-
val, sensitivity for signal detection is enhanced by the
factor <1.8.
The experimental conditions relevant to the data of
Fig. 2 are as follows. The incident local-oscillator
power is 6.4 X 10-4 W, the detector quantum efficien-
cies a f 0.89, and the shot noise is in excess of amplifi-
er noise by 20 dB at the modulation frequency Qo/27r =
1.6 MHz. The power P 2 = 8 X 10-8 W, while 6 = 8.5 X
10-4, 01/sin 0o = 0.015, and T > 0.98. The detection
bandwidth is 1.0 X 105 /sec (Gaussian filter) with two
postdetection video filters of time constants (10- sec,
5 X 10-4 sec). The signal-to-noise ratio evidenced in
Fig. 2(a) allows us to infer a modulation index 6' = 8.5
X 10-4, in excellent agreement with the measured
modulation index 6.
We should note that for the particular arrangement
that we have employed, the sensitivity could be greatly
enhanced by simply removing ml to increase the field
(sI) and hence the amplitude of the amplitude-mod-
ulation sidebands. Indeed, the case of squeezed light
in the configuration of Fig. 1 can only improve the
sensitivity to the level associated with (Eo) alone.
However, there are many applications (especially with
regard to spectroscopy) for which there are natural
upper limits to the fields that can be used to illuminate
various samples and for which the incident laser is
attenuated to provide a probe beam precisely in the
fashion of ml. If such attenuation is not permitted,
the squeezed vacuum beam P and the coherent field
.o can be combined not at a simple mirror but rather
with an optical cavity. The linewidth K of the cavity
would be chosen to be small compared with that asso-
ciated with S(Q). With the cavity locked at reso-
nance, the fields would be combined with .0 transmit-
ted through the cavity and Ps reflected from the cavity
output mirror. A squeezed coherent beam of approxi-
mate amplitude (Poy with a spectrum of squeezing
given by S(Q) could then be achieved, except for a loss
of squeezing around Q = 0 of width K. That the optical
parametric oscillator below threshold produces a
squeezed vaccum state and not a squeezed coherent
state is thus of little consequence.
In conclusion, we have demonstrated an improve-
ment in sensitivity of 2.5 dB for the measurement of
amplitude modulation beyond the limit set by the
vacuum-state or zero-point fluctuations of the electro-
magnetic field. Although the arrangement employed
may seem somewhat artificial relative to an actual
spectroscopic application, we note that the replace-
ment of the AOM by a sample whose, absorption is
externally modulated at Qo would provide a workable
squeezed-state spectrometer. In addition, the mea-
surement that we have described can be transcribed
into the language of frequency-modulation spectros-
copy by the simple association 01 sin 00 = Mg, where M
is the modulation index of the frequency-modulated
sidebands and u the differential sideband absorption.1
We gratefully acknowledge the contributions of J. L.
Hall. This research was supported by the U.S. Office
of Naval Research and by the Venture Research Unit
of British Petroleum America.
References
1. J. S. Snyder and R. A. Keller, eds., Ultrasensitive Laser
Spectroscopy, J. Opt. Soc. Am. B 2, 1428-1593 (1985).
2. H. J. Kimble and D. F. Walls, eds., Squeezed States of
the Electromagnetic Field, J. Opt. Soc. Am. B 4, 1450-
1741 (1987).
3. M. Xiao, L.-A. Wu, and H. J. Kimble, Phys. Rev. Lett.
59, 278 (1987).
4. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta,
Phys. Rev. Lett. 59, 2153 (1987).
5. L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys.
Rev. Lett. 57, 2520 (1986).
6. L.-A. Wu, M. Xiao, and H. J. Kimble, J. Opt. Soc. Am. B
4, 1465 (1987).
7. We denote Hilbert-space operators with a caret ().
8. A. Yariv, Optical Electronics (Holt, Rinehart, and Win-
ston, New York, 1985), Chap. 12.
9. H. P. Yuen and V. W. S. Chan, Opt. Lett. 8, 177 (1983).
10. M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386
(1984).


Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit

476 OPTICS LETTERS / Vol. 13, No. 6 / June 1988
Detection of amplitude modulation with squeezed light for
sensitivity beyond the shot-noise limit
Min Xiao, Ling-An Wu, and H. J. Kimble
Department of Physics, University of Texas at Austin, Austin, Texas 78712
Received January 22, 1988; accepted March 21, 1988
An improvement in precision beyond the limit set by the vacuum-state or zero-point fluctuations of the electromag-
netic field is reported for the detection of amplitude modulation encoded on a weak signal beam. The improvement
is achieved by employing the squeezed light from an optical parametric oscillator to reduce the level of fluctuations
below the shot-noise limit. An increase in signal-to-noise ratio of 2.5 dB relative to the shot-noise limit is
demonstrated.
An exploration of the limits on the detection of small
changes in amplitude of the electromagnetic field is of
fundamental and practical importance in optical
physics. While a variety of noise sources of technical
origin often limit sensitivity, the fundamental limit on
the detection of a small absorption or loss y has been
the so-called shot-noise limit (SNL), which sets a low-
er bound on the minimum detectable change 5A of the
coherent amplitude A of the field given by -y _ 6A/A
1//N, where N is the number of photons recorded in
the time interval of the measurement. A diverse set of
experiments in ultrasensitive laser spectroscopy1 pro-
vides but one example in which it has been possible to
overcome technical noise, to eliminate extraneous co-
herent signals, and to reach the SNL. However, with
conventional laser light in a coherent state, it is not
possible to proceed further to a sensitivity greater
than that specified by the SNL.
Improvements in precision beyond the SNL can be
realized by employing squeezed states of light. A
squeezed state is characterized by a phase-dependent
redistribution of quantum fluctuations such that the
variance of one of two quadrature components of the
field is reduced below that of the vacuum state. By
encoding the signal (e.g., amplitude modulation) on
the field variable with reduced fluctuations and by
employing a detection scheme that is largely insensi-
tive to the increased fluctuations of the conjugate field
variable, sensitivity greater than the SNL can be
achieved.3' 4
In this Letter we report experiments that utilize
squeezed light to improve the sensitivity for the detec-
tion of amplitude modulation by 2.5 dB relative to the
shot-noise or vacuum-state limit. In the measure-
ments a weak signal beam is combined with either a
vacuum-state or a squeezed-state input; the amplitude
of the resulting beam is modulated in propagation
through an acousto-optic modulator. The observed
increase in sensitivity for the detection of the signal
modulation is currently limited principally by simple
linear losses in propagation and detection and not by
the degree of squeezing available from our source.5' 6
Our apparatus is depicted in general outline in Fig.
1. Consider first the case of the inputs to m1 being a
coherent state -o and a vacuum state P,, with the
transmissivity of m1 being small. 7 The weak probe
beam t1 that is output from m1 is modulated in ampli-
tude at frequency go by an acousto-optic modulator
(AOM) to generate a field t 2 given by8
(A) = (E1 ) T COs 0(t) (1)
where 0(t) = So + 01 COS Qot and is proportional to the
amplitude of the index of refraction variation pro-
duced by the traveling acoustic wave in the modulator.
T is the static transmission coefficient (0 = 0) of the
AOM. For the case of low-loss operation of the modu-
lator we expand Eq. (1), keeping only lowest-order
terms at the modulation frequency Qo to find that
(E 2 ) _ (E 1 ) T(1 - 6 cos Q00, (2)
where the modulation index 6 = 01 sin 00. The modu-
lated field 22 enters the signal port of a balanced
homodyne detector formed by the beam splitter m2
OPO A
, , Es
A
ELO'
E0
Di
12
Fig. 1. Schematic of the apparatus for detection of ampli-
tude modulation with squeezed states. The reflectivity of
ml is close to unity; m 2 is a beam splitter with R = T = 0.5.
0146-9592/88/060476-03$2.00/0 © 1988, Optical Society of America
n -- |I AOM
June 1988 / Vol. 13, No. 6 / OPTICS LETTERS 477
(50-50) and the photodiodes D1 and D2 and operated
such thatI(PLO)I» I(A12)I.9 The amplified photocur-
rents il and i2 from the two arms are combined with a
1800 phase shift to produce the photocurrent i, which
is directed to a commercial spectrum analyzer.
The photocurrent i contains a signal component is at
frequency 9o of rms amplitude
i,(Qo) = 2Tea- (-PLO) (EA) 6 COS3, (3)
where the units of the fields are chosen such that(P.t) expresses a flux in photons per second across
the detector area. The quantum efficiency of the pho-
todiodes is given by ar, while X expresses the hetero-
dyne efficiency for overlap of the fields E2 and tLO- 3
is the phase between (E2 ) and (BLO). For d = +=
pwr (p an integer), the fields are in phase, and the
amplitude modulation encoded on E2 results in ampli-
tude modulation of ELO. For d = 0_ = (2p + 1)r/2,
the modulation of P2 results in phase modulation of
BLO and hence produces no signal current to lowest
order in 6. In addition to the signal current i,(Q0),
there is a noise current in against which any coherent
detection must be made. In the case of a vacuum-
state input for ES, in is just the shot-noise current of
rms amplitude
in = (2eiB)1 12 = (2e2 a (ELOttLO) B)1 2 . (4)
T'(Q0o, ) = TI/[1 + AS(Qo, ct)] = cP262 cos 2 0 (6)
where r represents the overall efficiency for propaga-
tion and detection of the squeezed light.6 The two
phases (S, O) appearing in Eq. (6) specify, respectively,
the orientation of the coherent modulation at jo and
the orientation of the squeezed-noise ellipse relative to
the local-oscillator field. For 0 = 0+ = q7r (q an inte-
ger), the enhanced fluctuations S+ of the squeezed
state align with (PLO), while for 0 = 0- = (2q + 1)ir/2
the reduced fluctuations S_ of the squeezed state align
with (ELO). From Eq. (6) we see that for 0 = 0-, an
improvement in signal-to-noise ratio beyond the shot-
noise or vacuum-state limit is given by the factor (1 +
PS_)-1>> 1 for S& -1 and v 1. On the other hand,
for 0 = 0+, there is a large degradation in the signal-to-
noise ratio by the factor (1 + PS+)-1 << 1, pointing out
the requirement for precise control of 0.
D(dB)
0The signal-to-noise ratio for coherent-state inputs (P0,
PLO) and a vacuum-state input P, is thus
TV = is2 aP 2 cos 2 2i2 B (5)
where we have defined the effective power P2Tnq2I(_ 1)j2 (photons/sec). For ,3 = 3+, a signal-to-
noise ratio of unity (TV = 1) implies a minimum de-
tectable modulation index 6 = (B/aP 2)1 /2 = 11VN2,
with N2 as the mean number of photoelectrons in the
measurement interval B-1 due to P2. This limit on the
minimum detectable amplitude modulation is the
SNL discussed above with the association of 6 with y.
Any improvement beyond this limit is not possible in
conventional detection schemes with coherent states.
An improvement in sensitivity is possible, however,
if a squeezed field ES is injected in place of the vacuum
field at m1 to reduce the noise against which the modu-
lation signal must be detected. The squeezed light in
our experiment is generated by a subthreshold optical
parametric oscillator with degenerate signal and idler
waves at X = 1.064 Am. 5 ,6 The output field PL from the
optical parametric oscillator is characterized by a
spectrum of squeezing S(Q, q), where Q is the frequen-
cy offset from the carrier and p selects the quadrature
phase. 10 S(Q2, k) is normalized such that S = 0 for a
vacuum state, while for a perfectly squeezed field, S -
-1 for one quadrature and S -- for the conjugate
quadrature. With the injection of squeezed vacuum
at m1 and under the assumptions of modest squeezing
and small modulation index, the calculation of the
signal current proceeds exactly as before [Eq. (3)].
However, the noise in the difference photocurrent now
becomes in2(Q, ck) = in 2[1 + AS(Q, 1)], resulting in a
signal-to-noise ratio
D (d B)
A= 7r Phase :
Fig. 2. Spectral density 4? of photocurrent fluctuations ver-
sus signal phase 3 for fixed analysis frequency Qo/27r = 1.6
MHz. The phase f between (E2) and (ELO) is swept in a
sawtooth fashion to produce the periodic variation in 4? with
the sharp features corresponding to the flyback to restart
the scan. The dashed line gives the vacuum level obtained
with inputs E0 and ES blocked. The gap in (a) is caused by a
trigger event that interrupts the local-oscillator beam. (a)
Vacuum-state input for E5, (b) squeezed-state input for E5.The slow variation of (D over the scan in (b) results from a
variation in 0 to adjust for minimum noise level. An im-
provement of 2.5 dB in signal-to-noise ratio is achieved in (b)
relative to (a) near the noise minimum. The time for the
entire trace is 0.2 sec.
478 OPTICS LETTERS / Vol. 13, No. 6 / June 1988
Our experimental results for the improvement in
sensitivity for the detection of amplitude modulation
are displayed in Fig. 2, which records the spectral
density 4? of photocurrent fluctuations of the differ-
ence photocurrent i as a function of phase /. The
periodic up-down variation of 4? with / reflects the cos /
dependence of the signal photocurrent as given by Eq.
(3). For : = A+, the signal modulation is detected with
maximum efficiency (cos2 d = 1), and the detected
level 4' is the sum of contributions from the coherent
modulation and the noise current. For / = A-, the
modulation is predominantly frequency modulation
with respect to the local-oscillator field, and no coher-
ent signal is expected (cos2 / = 0). Hence the detected
level in this situation should correspond to photocur-
rent noise alone, as given by 4? = 10 log (1 + CS). In
Fig. 2(a) the input A, to ml is a vacuum state for which
S = 0, while in Fig. 2(b) a squeezed state is injected for
which S < 0 for 1 properly chosen. The actual vacu-
um-state or shot-noise level in Fig. 2 is indicated by a
dashed line and is obtained with the inputs A, and Po
blocked after the procedure discussed in Ref. 6. Fig-
ure 2(a) corresponds roughly to the case T'I 1 (signal-
to-noise ratio of unity) and represents the SNL for
signal detection. A decrease in noise level and an
increase in signal-to-noise ratio beyond this limit are
shown in Fig. 2(b). Figures 2(a) and 2(b) are recorded
under identical conditions with the single exception
that squeezed-vacuum light is injected into ml for Fig.
2(b). In Fig. 2(b) the angle 0 is slowly varied to reach
a position of minimum noise. On the left-hand edge
of the figure the noise level is well in excess of the SNL.
As a value near 1 = 0. is reached in the center of the
figure, there is a decrease in photocurrent noise below
the SNL. From a number of traces as in Fig. 2, we find
an improvement in signal-to-noise ratio relative to the
vacuum-state limit of (2.5 t 0.5) dB (a factor of 1.8 in
terms of noise power). The consequence of this im-
provement is that a given level of precision for i,(Qo)
could be reached in a measurement time shortened by
a factor <1.8 relative to shot-noise-limited measure-
ments. Alternatively, for a fixed measurement inter-
val, sensitivity for signal detection is enhanced by the
factor <1.8.
The experimental conditions relevant to the data of
Fig. 2 are as follows. The incident local-oscillator
power is 6.4 X 10-4 W, the detector quantum efficien-
cies a f 0.89, and the shot noise is in excess of amplifi-
er noise by 20 dB at the modulation frequency Qo/27r =
1.6 MHz. The power P 2 = 8 X 10-8 W, while 6 = 8.5 X
10-4, 01/sin 0o = 0.015, and T > 0.98. The detection
bandwidth is 1.0 X 105 /sec (Gaussian filter) with two
postdetection video filters of time constants (10- sec,
5 X 10-4 sec). The signal-to-noise ratio evidenced in
Fig. 2(a) allows us to infer a modulation index 6' = 8.5
X 10-4, in excellent agreement with the measured
modulation index 6.
We should note that for the particular arrangement
that we have employed, the sensitivity could be greatly
enhanced by simply removing ml to increase the field
(sI) and hence the amplitude of the amplitude-mod-
ulation sidebands. Indeed, the case of squeezed light
in the configuration of Fig. 1 can only improve the
sensitivity to the level associated with (Eo) alone.
However, there are many applications (especially with
regard to spectroscopy) for which there are natural
upper limits to the fields that can be used to illuminate
various samples and for which the incident laser is
attenuated to provide a probe beam precisely in the
fashion of ml. If such attenuation is not permitted,
the squeezed vacuum beam P and the coherent field
.o can be combined not at a simple mirror but rather
with an optical cavity. The linewidth K of the cavity
would be chosen to be small compared with that asso-
ciated with S(Q). With the cavity locked at reso-
nance, the fields would be combined with .0 transmit-
ted through the cavity and Ps reflected from the cavity
output mirror. A squeezed coherent beam of approxi-
mate amplitude (Poy with a spectrum of squeezing
given by S(Q) could then be achieved, except for a loss
of squeezing around Q = 0 of width K. That the optical
parametric oscillator below threshold produces a
squeezed vaccum state and not a squeezed coherent
state is thus of little consequence.
In conclusion, we have demonstrated an improve-
ment in sensitivity of 2.5 dB for the measurement of
amplitude modulation beyond the limit set by the
vacuum-state or zero-point fluctuations of the electro-
magnetic field. Although the arrangement employed
may seem somewhat artificial relative to an actual
spectroscopic application, we note that the replace-
ment of the AOM by a sample whose, absorption is
externally modulated at Qo would provide a workable
squeezed-state spectrometer. In addition, the mea-
surement that we have described can be transcribed
into the language of frequency-modulation spectros-
copy by the simple association 01 sin 00 = Mg, where M
is the modulation index of the frequency-modulated
sidebands and u the differential sideband absorption.1
We gratefully acknowledge the contributions of J. L.
Hall. This research was supported by the U.S. Office
of Naval Research and by the Venture Research Unit
of British Petroleum America.
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4. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta,
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7. We denote Hilbert-space operators with a caret ().
8. A. Yariv, Optical Electronics (Holt, Rinehart, and Win-
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eds., Ultrasensitive Laser Spectroscopy,

Optical Electronics

We denote Hilbert-space operators with a caret ().

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Detection of amplitude modulation with squeezed light for sensitivity beyond the shot-noise limit

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