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Observation of sub-poissonian photoelectron statistics in a negative feedback semiconductor laser
Volume 57, number 4
OPTICS COMMUNICATIONS
15 March 1986
OBSERVATION OF SUB-POISSONIAN PttOTOELECTRON STATISTICS IN A NEGATIVE FEEDBACK S E M I C O N D U C T O R LASER S. M A C H I D A and Y. Y A M A M O T O \ T T Mu.~avhim: t;'[ectrica: ( ommumcatirm l.ahoramrie~, :~h~.~a.~him~-~hi. "Fokv~ ]~¢(L Japan
Received 30 JuG 1985: rcxised manuscript received 24 September 1985
"lhe photo-current fluctuation spectra and the photoelectron Matistics of a negative fccdback stabilized semiconductor laser are measured. The pht~to-currcrlt spectral density is reduced b,, 7 dB below the standard quantum noise level, 2e/rJ Sub-poissonian phou~electron s|atistics with variance ( 2~n 2 ) 0.26 'i n ) are observed.
Recently, considerable efforts are made to generate the nonclassical photons such as photon number state, sub-poissonian state and squeezed state [1 4]. The phenomena related to the nonclassical photons, such as sub-poissonian photon statistics, photon antibunching and squeezing, cannot be described by the classical theory. Mathematically, this corresponds to the fact that these states do not have a positive nonsingular representation in terms of the GlauberSudarshan P ( a ) distribution [5,6]. There is hope that nonclassical photons will improve the signal-to-noise ratio in any optical precision measurement. An optical interferometer gravitational wave detector [7] and an optical homodyne/heterodyne communication [8] with a squeezed state, and a photon counting communication with a sub-poissonian state are candidates. Although the squeezing phenomenon has not yet been observed, weak sub-poissonian photon statistics were observed in resonance fluorescence [9] and in Franck-Hertz light [ 10]. A new approach to generating such a sub-poissonJan state was proposed, in which the photon flux fluctuation of the laser output field is measured and an electrical negative feedback circuit is employed to suppress it [t 1]. The proposed scheme was theoretically studied by using the quantum mechanical Langevin equations [ t 2 ], wtfich is briefly summarized here. According to the quantum photodetection theory [8], an ideal photon detector with unity quart290
tum efficiency "measures" the quantum operator, = e r ~ r, where/t = r Tr is a photon flux operator of the laser output field. Although the fluctuationin the photon detector output current is a classical random variable it keeps the quantum stochastic properties of the incident laser field. This is the key point in understanding the operational principle of the proposed scheme. Quantum mechanical negative correlation between the photon flux fluctuation and the counter-modulation of a laser pumping rate is thus established to suppress the quantum noise, if the phase reversed photon detector output current is added to the dc bias current of the laser. In order to clarify the difference between the conventional open loop modulation of a classical light and the closed loop negative feedback scheme, we mention the famous Franck-Hertz effect [13]. The electron emission process in a space charge limited vacuum tube exhibits sub-poissonian statistics. When the electron emission rate increases above its average rate, the potential profile between the cathode and the anode is deformed due to the Coulomb repulsion to decrease the emission rate, and vice versa. Indeed, the sub-poissonian light was generated by using such Franck-Hertz electrons as a pump source in the optical excitation [10]. There is a certain analogy between the present scheme and this Franck-Hertz light. The Coulomb repulsion establishes the negative correlation between the electron emission rate and the potential profile. In the present scheme, this natural 0 0304018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 57, number 4
(a)
bias
OPTICS COMMUNICATIONS
'-4 Feedback
[
Measurement
Loop
Circuit c I
DB
r---__ r--Q) B
LD
(b)
G
©A Fig. 1. Configuration of the experimental set up with a single photodetector (a) and with a beam splitter/dual photodetector (b).
feedback mechanism is replaced by the artificial electrical circuit. A similar negative feedback for the laser frequency fluctuation was demonstrated to reduce the spectral linewidth below the modified Schawtow-Townes limit [14]. The quantum mechanical Langevin analysis explains the experimental result successfully [ 15]. The experimental setup is shown in fig. l(a). A single longitudinal mode GaAs semiconductor laser (HLP-1400) having a 0.82/~m oscillation wavelength was used at a bias level ofI/Ith = 2--3. The laser output was detected by a Si photodiode having a quantum efficiency larger than 0.8. The phase-reversed and amplified photocurrent fluctuation is superposed on a d c bias current to counteract the laser output photon flux fluctuation. The photo-current fluctuation was connected to the noise spectrum measurement circuit and to the photoelectron statistics measurement circuit. The photo-current fluctuation spectrum was measured with a spectrum analyzer (HP-8568A). The photoelectron statistics were measured using the analog photon counting technique [16] with the integrator (LPF; Anritsu MN51A), the sampling oscilloscope (Tektronix 7904-s2), and the digital wave memory (Kikusui 87025) having 50 to 64 channels. The data number
15 March 1986
for each photoelectron counting measurement is 3 × 105 . Another experimental set up is shown in fig. l(b). The laser output was equally divided by a 50-50% beam splitter to illuminate two identical Si photodiodes having a quantum efficiency larger than 0.8. The five differential amplifiers were carefully chosen so as to keep the overall gains from the photodiodes to the three output terminals, A, B and C, exactly equal and flat over the measurement bandwidth up to 15 MHz. The photo-current fluctuation at terminal A was negatively fed back to the injection current of the laser diode in a similar way to that in fig. l(a). The quantum limited photo-current spectral density and the poissonian photoelectron statistics were calibrated as the absolute level. Identifying the absolute quantum noise levels in experiments is difficult, because the Si photodiode quantum efficiency is not constant. Rather it indicates a dependence on the fluctuation frequency. It is additionally difficult because the absolute power level displayed on the spectrum analyzer is not reliable. Furthermore, identifying the poissonian photoelectron statistics is complex because a conventional digital photon counter having a threshold decision is incapable of handling large number of photoelectrons on the order of 108 as in the present case. In order to overcome such difficulties, an incoherent GaAs light emitting diode (LED) output was used as a quantum noise generator. The measurement time interval T ~ 10 - 7 - 1 0 -8 s in our experiment is much longer than a coherent time, rc, which is on the order of 10 -12 s for a GaAs LED. The LED light therefore produces the quantum limited photo-cur. rent spectrum and the poissonian photoelectron statistics. The measured photo-current spectrum are compared with the quantum noise level, 2eIp, where Ip is the average photocurrent. The photo-current noise power spectrum, Pin(g~), measured with the spectrum analyzer is transformed into the primary current noise spectrum by Pin(~2)/GB, where G is the measured electronic amplifier circuit power gain and B is the separately calibrated resolution bandwidth of the spectrum analyzer. It is further transformed into the excess noise factor, ×1, defined as the photo-current spectral density divided by the quantum noise level as X1 = Pin (~)[2eIpGB-
(1) 291
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The thermal noise added by the electronic amplifier was 1 0 - 1 5 dB lower than the quantum noise level and it was removed b y a light chopper and the phase sensitive detection using a lock-in amplifier. The experimental result deviates from the theoretical value, X = 1, only by less than 0.05 dB in the frequency region from dc to 15 MHz. This can be considered as the accuracy o f the present photo-current spectrum measurement. The measured and theoretical poissonian photoelectron statistics are compared. The integrated photocurrent, A = ftt+Ti(t)dt over the measurement time interval, T, is transformed into the primary photoelectron number, n, by the relation n = A/ev~.. The theoretical poissonian distribution is calculated by P(n) = e x p ( - ( n ) ) (n)n/n !, where (n) is the average number o f primary photoelectrons and is given by (n) = lpT/e. The agreement between the experimental and theoretical poissonian statistics is fairly good. The slight deviation is due to thermal noise contribution, which was excluded in the following way. If the normalized second m o m e n t , X, of a photoelectron number is defined as X ~- [((z2~'/)2) - (n)]/(n},
(2)
then X is related to the excess intensity noise factor, X2, as
15 March 1986
X2 - 1 = X.
(3)
We measured a zero mean thermal noise electron distribution b y blocking the incident LED light and obtained the second moment o f the thermal noise electron, ((2~n)2)th . The primary photoelectron second m o m e n t , ( ( ~ ) 2 } , is obtained by subtracting the thermal noise contribution: ((ZX/'/)2} = Zn2p(n) -D2nP(n)] 2 - ((2xn)2}th .
When the photoelectron statistics, P(n), is poissonian, the value o f X is zero, which corresponds to X2 = 1. Table 1 summarizes the values o f Xl obtained from the photo-current spectrum measurement, using (1), and X2 from the photoelectron counting measurement, using (3) respectively. As can be seen, the experimental results are very close to the quantum limit of X = 1.0. The photo-current fluctuation spectra after the low pass filtering for the free-running and feedback stabilized GaAs laser output (fig. l(a)) are shown in fig. 2(a) with the calibrated quantum noise level produced by an LED light. While the photo-current fluctuation spectrum for the free-running condition is higher than the quantum noise level b y 4 - 5 dB, that for the feedback stabilization is reduced below this noise level b y 6 - 7 dB, depending on the frequency. The photoelectron statistics for the free-running and
Table 1 The excess intensity noise factors, xl, obtained from the photo-current fluctuation spectrum measurement, (1), and x2 from the photo-electron counting measurement, (3), for the three quantum noise level calibration cases using an LED light, a freerunning semiconductor laser and a feedback stabilized semiconductor laser. Detector bandwidth (Measurement time interval)
Ip
6.713 MHz (74.48 ns) 9.463 MHz (52.84 ns) 13.10 MHz (38.17 ns)
1.10 mA
Free-running GaAs laser
13.10 MHz (38.17 ns)
1.10 mA
2.747 (4.388 dB)
2.621 × 108
2.821 (4.504 dB)
Feedbackstabilized GaAs laser
13.10 MHz (38.17 ns)
1.10 mA
0.207 (-6.848 dB)
2.621 X 108
0.258 (-5.887 riB)
Calibration with a LED
"~ "3 ~9~-
1.10 mA 1.10 mA
(4)
xl (Photo-current fluctuation spectrum) 1.005 (0.02 dB) 0.9954 (-0.02 dB) 0.9908 (-0.04 dB)
{n)
x2 (Photoelectron counting)
5.114 X 108
1.0002 (0.001 dB) 1.003 (0.013 dB) 1.002 (0.008 dB)
3.628 X 108 2.621 X 108
Volume 57, number 4
OPTICS COMMUNICATIONS
15 March 1986
(a)
Free-running LD rn
0
o,1
-5
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LED
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, 5
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Fig. 2. (a) Normalized photo-current fluctuation spectral densities for free-running and negative feedback semiconductor laser having a single photodetector. (b) Photoelectron statistics for free-running and negative feedback semiconductor laser with a single photodetector. feedback stabilized GaAs laser output (fig. 1(a)) are compared in fig. 2(b) with the theoretical poissonian distribution. While the photoelectron statistics for the free-running condition are super-poissonian, the sub-poissonian statistics are obtained during the feedback stabilization. The excess noise factor, Xl, obtained from the photo-current fluctuation spectrum and X2, from the photoelectron statistics, are shown in table 1 for the free-running and feedback stabilization conditions. It is obvious that the observed sub-poissonian photoelectron statistics and sub-quantum limit photo-current fluctuation spectrum are much more dominant phenomena as opposed to the measurement uncertainty calibrated by using a LED light. Since the photoelectron emission process is a one-to-one correspondence with the photon arrival rate when the detector quantum efficiency is high [8], the above experi-
mental results cannot be explained without assuming that the GaAs laser output features the sub-poissonian photon statistics. The photo-current fluctuation spectra measured at terminals A and B for the free-running GaAs/ A1GaAs semiconductor laser (fig. 1(b)) are shown by the circles and plusses in fig. 3 and compared with the theoretical quantum noise level shown by a solid line. The incident waves on the two Si photodiodes exhibit smaller excess noise than the result of the single detector shown in fig. 2(a). This is because the semiconductor laser is biased higher and also because there is optical loss present between the laser and the photodiode, which contributes to the elimination of the excess noise of the incident wave. When the feedback circuit was closed, the noise spectrum at terminal A was reduced by 5 - 1 0 dB below the quantum noise level. The difference is again 293
Volume 57, number 4
OPTICS COIVlMUNICATIONS
15 March 1986
A
m
5
-~ •-~
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.g~
without feedback
°
(C)
,~with
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i
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_10~
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20
Frequency (MHz)
Fig. 3. (A) The current fluctuation spectra at terminal A. A and B correspond to the cases with and without feedback stabilization. Vuenr~isch~tS~seSledeB[~{vt2°~Z°~st~ 2~dleB 5+M+tl:/edrlmVin(B'TJmoCoU:r:;ntiflnU~t~, ~°Xnxs:PteCtr;a~! 'n~q?on~d ~ i ° ; 2 d f i Z bYst~edqu~n represent the quantum noise levels for terminals A, B and for terminal C.
much larger than the uncertainty o f less than 0.05 dB in identifying the quantum noise level. This means that the incident wave on photodiode A has its reduced noise below the standard quantum limit. On the other hand, the noise spectrum at terminal B was increased by about 3 dB above the quantum noise level. This means that the incident wave on photodiode B has its noise increased to twice that of the quantum noise level. The photoelectron counting measurement exhibits the same behavior. The photoelectron statistics measured at terminal A for free-running and feedback stabilized cases are shown by the crosses and open circles in fig. 4. The theoretical poissonian distribution is shown by a solid line. It is clear in fig. 4 that the super-poissonian photoelectron statistics of a free running laser output become sub-poissonian through feedback stabilization. The experimental photoelectron statistics give the normalized second moment X = 0.81 in a feedback stabilization case, which corresponds to the excess noise factor X = 0.19 ( 7 dB). This value is in fairly good agreement with the reduced current noise spectrum shown in fig. 3. The photon statistics measured at terminal B, on the other 294
hand, exhibit a super-poissonian distribution for feedback stabilization which is broader than that for the free-running case. Again the measured value o f X = +0.87 (or X = 1.87) is in fairly good agreement with the enhanced current noise spectrum shown in fig. 3. These experimental results can be understood as follows: If the output wave from the semiconductor laser and the zero-point fluctuation incident on a beam splitter from an open port are designated by t:1 and ~, the incidental waves on photodiodes A and B are given by
~A = (il + 0)Ix/E,
(5)
and iB = (#1 -- ()/N/fT"
(6)
Here, rl (= ro + ZX#l)exp(-i2x7)) and the zero point fluctuation 0 ( - 01 + ib2) satisfy (O1) = (02) = O.
(7)
If the photodiode quantum efficiency is unity, a d c photocurrent, iA0, is given by iA0 ----e{#Af A) = er2/2"
(8)
Volume 57, number 4
OPTICS COMMUNICATIONS
15 March 1986
0
(A)
o~ o
Xx=0.26
6x10 I
Xo = - 0 . 8 1 0 0
4x1() 5
0
%
o
2x18l i oL
-2&n
-An
0 n-
&n
2&n
i
I 8x 1()5 ~
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4x 1~s~ I
Xx=0.31 Xo=0.87
(B)
2x 1()5r li o
~~,~o
~ o~.,~ "~ / -2An -An
0 n-
", ~oo
~£~o An 2An
Fig. 4. The photoelectron probabilities at terminals A and B with and without feedback stabilization. The integration time interval is T = 37 ns. The solid lines are theoretical Poisson distributions for the average photoelectron number (n) = 1.5 X 108 . XXX : without feedback, ooo: with feedback. An = 2 × 104. The p h o t o c u r r e n t fluctuation, A/A, o f p h o t o d i o d e A "measures" the i n t e n s i t y f l u c t u a t i o n operator [8] as
A/B, o f the p h o t o d i o d e B "measures" A/'B = 2 e ( / B 1 ) ~ B 1 = e r 0 ( z ~ l -- Cl)-
A/A = 2 e ( i a l ) Z~A1 = ero (2g:l + Cl )-
(9)
The q u a n t u m mechanical correlation established b y feedback stabilization is
~1 = ~1-
(]o)
Consequently, the fluctuation spectra of the incident wave, i A , on p h o t o d i o d e A is reduced. On the other hand, the photo-current fluctuation,
(1 1)
Eq. (1 1) explains the e n h a n c e d i n t e n s i t y noise spectral density and super-poissonian p h o t o e l e c t r o n statistics. In order to confirm the above discussion, the photo-current f l u c t u a t i o n spectral density and photoelectron statistics were measured at terminal C, that is, the subtract o u t p u t . The dc p h o t o c u r r e n t at terminal C is expressed as 295
Volume 57, number 4
OPTICS COMMUNICATIONS
ic0 = e(<#tA#A) -- (/:B~/:B))= e(r 2 -- r02) = 0.
(12)
15 March 1986
Fmuio. Kanaya of Musashino ECL, NTT for their useful discussions.
The photocurrent fluctuation, Aic, "measures" the following operator [17] 3Jc = 2e(<#A1>2~A1 -- (#nl > ~ B 1 ) = 2er0?l •
(13)
Eq. (13) suggests that the photocurrent fluctuation, Aic, measured at terminal C reflects the beat noise between the coherent excitation of the laser output wave, r0, and the in-phase component of the zero-point fluctuation, kl" It further suggests that it is always standard quantum-limited irrespective of the feedback stabilization. This was actually confirmed by the experiment. The photo-current fluctuation spectral densities measured at terminal C for freerunning and feedback stabilization cases are shown by the crosses and triangles in fig. 3, which are just 3 dB above the q u a n t u m noise level for terminals A and B. Although the sub-poissonian state generated inside a negative feedback loop cannot be extracted by a conventional output coupling scheme as described above, extraction is possible if the q u a n t u m nondemolition (QND) measurement [18,19] is used instead of the conventional beam splitter and photodetector to measure the photon flux noise of the laser output field. The QND detector can measure the q u a n t u m state without perturbing (destroying) it. Indeed, the QND measurement of a photon number is possible via optical Kerr effect [20]. The combination of the negative feedback semiconductor laser and the QND measurement of a photon number can generate the subpoissonian state observed here outside a feedback loop [12].
References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11 | [12] [13] [14] [15] [16]
[17] [18] [19]
The authors wish to thank Prof. Hermann A. Haus of Massachusetts Institute of Technology, Prof. Olle Nilsson of the Royal Institute of Technology and Dr.
296
[20]
H. Takahashi, Adv. Commun. Syst. 1 (1965) 227. D. Stoler, Phys. Rev. D4 (1971) 1925. H.P. Yuen, Phys. Rev. A13 (1976) 2226. F. Milburn and D.F. Wails, Optics Comm. 39 (1981) 401. R.J. Glauber, Phys. Rev. 131 (1963) 2766. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277. C.M. Caves, Phys. Rev. D23 (1981) 1693. H.P. Yuen and J.H. Shapiro, IEEE Trans. Inform. Theory, IT-26 (1980) 78. R. Short and L. Mandel, Phys. Rev. Lett. 51 (1983) 384. M.C. Teich, B.E.A. Saleh and D. Stoler, Optics Comm. 46 (1983) 244; M.C. Teich and B.E.A. Saleh, J. Opt. Soc. Am. B2 (1985) 275. Y. Yamamoto, O. Nilsson and S. Saito, Proc. of 3rd USJapan Seminar on Coherence, Incoherence and Optical Chaos in Quantum Electronics, 1984, Nara, Japan, p. 7. Y. Yamamoto, N. Imoto and S. Machida, Phys. Rev. A., to be published. J. Franck and G. Hertz, Vet. Dtsch. Phys. Ges. 16 (1914)457. S. Saito, O. Nilsson and Y. Yamamoto, Appl. Phys. Lett. 46 (1985) 3. Y. Yamamoyo, O. Nilsson and S. Saito, IEEE J. Quantum Electron. QE-21 (1985) 1919. P.L. Liu, L.E. Fencil, J.S. Ko, I.P. Kaminow, T.P. Lee and C.A. Burrus, IEEE J. Quantum Electron. QE-19 (1983) 1348. H.P. Yuen and V.W.S. Chart, Optics Lett. 8 (1983) 177. V.B. Braginsky, Y.I. Vorontsov and K.S. Thorne, Science 209 (1980) 547. C.M. Caves, K.S. Thorne, R.W.P. Drever, V.D. Sandberg, and M. Zilnmermann, Rev. Mod. Phys. 52 (1980) 341. N. Imoto, H.A. Haus and Y. Yamamoto, Phys. Rev. A32 (1985) 2287.
OPTICS COMMUNICATIONS
15 March 1986
OBSERVATION OF SUB-POISSONIAN PttOTOELECTRON STATISTICS IN A NEGATIVE FEEDBACK S E M I C O N D U C T O R LASER S. M A C H I D A and Y. Y A M A M O T O \ T T Mu.~avhim: t;'[ectrica: ( ommumcatirm l.ahoramrie~, :~h~.~a.~him~-~hi. "Fokv~ ]~¢(L Japan
Received 30 JuG 1985: rcxised manuscript received 24 September 1985
"lhe photo-current fluctuation spectra and the photoelectron Matistics of a negative fccdback stabilized semiconductor laser are measured. The pht~to-currcrlt spectral density is reduced b,, 7 dB below the standard quantum noise level, 2e/rJ Sub-poissonian phou~electron s|atistics with variance ( 2~n 2 ) 0.26 'i n ) are observed.
Recently, considerable efforts are made to generate the nonclassical photons such as photon number state, sub-poissonian state and squeezed state [1 4]. The phenomena related to the nonclassical photons, such as sub-poissonian photon statistics, photon antibunching and squeezing, cannot be described by the classical theory. Mathematically, this corresponds to the fact that these states do not have a positive nonsingular representation in terms of the GlauberSudarshan P ( a ) distribution [5,6]. There is hope that nonclassical photons will improve the signal-to-noise ratio in any optical precision measurement. An optical interferometer gravitational wave detector [7] and an optical homodyne/heterodyne communication [8] with a squeezed state, and a photon counting communication with a sub-poissonian state are candidates. Although the squeezing phenomenon has not yet been observed, weak sub-poissonian photon statistics were observed in resonance fluorescence [9] and in Franck-Hertz light [ 10]. A new approach to generating such a sub-poissonJan state was proposed, in which the photon flux fluctuation of the laser output field is measured and an electrical negative feedback circuit is employed to suppress it [t 1]. The proposed scheme was theoretically studied by using the quantum mechanical Langevin equations [ t 2 ], wtfich is briefly summarized here. According to the quantum photodetection theory [8], an ideal photon detector with unity quart290
tum efficiency "measures" the quantum operator, = e r ~ r, where/t = r Tr is a photon flux operator of the laser output field. Although the fluctuationin the photon detector output current is a classical random variable it keeps the quantum stochastic properties of the incident laser field. This is the key point in understanding the operational principle of the proposed scheme. Quantum mechanical negative correlation between the photon flux fluctuation and the counter-modulation of a laser pumping rate is thus established to suppress the quantum noise, if the phase reversed photon detector output current is added to the dc bias current of the laser. In order to clarify the difference between the conventional open loop modulation of a classical light and the closed loop negative feedback scheme, we mention the famous Franck-Hertz effect [13]. The electron emission process in a space charge limited vacuum tube exhibits sub-poissonian statistics. When the electron emission rate increases above its average rate, the potential profile between the cathode and the anode is deformed due to the Coulomb repulsion to decrease the emission rate, and vice versa. Indeed, the sub-poissonian light was generated by using such Franck-Hertz electrons as a pump source in the optical excitation [10]. There is a certain analogy between the present scheme and this Franck-Hertz light. The Coulomb repulsion establishes the negative correlation between the electron emission rate and the potential profile. In the present scheme, this natural 0 0304018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 57, number 4
(a)
bias
OPTICS COMMUNICATIONS
'-4 Feedback
[
Measurement
Loop
Circuit c I
DB
r---__ r--Q) B
LD
(b)
G
©A Fig. 1. Configuration of the experimental set up with a single photodetector (a) and with a beam splitter/dual photodetector (b).
feedback mechanism is replaced by the artificial electrical circuit. A similar negative feedback for the laser frequency fluctuation was demonstrated to reduce the spectral linewidth below the modified Schawtow-Townes limit [14]. The quantum mechanical Langevin analysis explains the experimental result successfully [ 15]. The experimental setup is shown in fig. l(a). A single longitudinal mode GaAs semiconductor laser (HLP-1400) having a 0.82/~m oscillation wavelength was used at a bias level ofI/Ith = 2--3. The laser output was detected by a Si photodiode having a quantum efficiency larger than 0.8. The phase-reversed and amplified photocurrent fluctuation is superposed on a d c bias current to counteract the laser output photon flux fluctuation. The photo-current fluctuation was connected to the noise spectrum measurement circuit and to the photoelectron statistics measurement circuit. The photo-current fluctuation spectrum was measured with a spectrum analyzer (HP-8568A). The photoelectron statistics were measured using the analog photon counting technique [16] with the integrator (LPF; Anritsu MN51A), the sampling oscilloscope (Tektronix 7904-s2), and the digital wave memory (Kikusui 87025) having 50 to 64 channels. The data number
15 March 1986
for each photoelectron counting measurement is 3 × 105 . Another experimental set up is shown in fig. l(b). The laser output was equally divided by a 50-50% beam splitter to illuminate two identical Si photodiodes having a quantum efficiency larger than 0.8. The five differential amplifiers were carefully chosen so as to keep the overall gains from the photodiodes to the three output terminals, A, B and C, exactly equal and flat over the measurement bandwidth up to 15 MHz. The photo-current fluctuation at terminal A was negatively fed back to the injection current of the laser diode in a similar way to that in fig. l(a). The quantum limited photo-current spectral density and the poissonian photoelectron statistics were calibrated as the absolute level. Identifying the absolute quantum noise levels in experiments is difficult, because the Si photodiode quantum efficiency is not constant. Rather it indicates a dependence on the fluctuation frequency. It is additionally difficult because the absolute power level displayed on the spectrum analyzer is not reliable. Furthermore, identifying the poissonian photoelectron statistics is complex because a conventional digital photon counter having a threshold decision is incapable of handling large number of photoelectrons on the order of 108 as in the present case. In order to overcome such difficulties, an incoherent GaAs light emitting diode (LED) output was used as a quantum noise generator. The measurement time interval T ~ 10 - 7 - 1 0 -8 s in our experiment is much longer than a coherent time, rc, which is on the order of 10 -12 s for a GaAs LED. The LED light therefore produces the quantum limited photo-cur. rent spectrum and the poissonian photoelectron statistics. The measured photo-current spectrum are compared with the quantum noise level, 2eIp, where Ip is the average photocurrent. The photo-current noise power spectrum, Pin(g~), measured with the spectrum analyzer is transformed into the primary current noise spectrum by Pin(~2)/GB, where G is the measured electronic amplifier circuit power gain and B is the separately calibrated resolution bandwidth of the spectrum analyzer. It is further transformed into the excess noise factor, ×1, defined as the photo-current spectral density divided by the quantum noise level as X1 = Pin (~)[2eIpGB-
(1) 291
Volume 57, number 4
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The thermal noise added by the electronic amplifier was 1 0 - 1 5 dB lower than the quantum noise level and it was removed b y a light chopper and the phase sensitive detection using a lock-in amplifier. The experimental result deviates from the theoretical value, X = 1, only by less than 0.05 dB in the frequency region from dc to 15 MHz. This can be considered as the accuracy o f the present photo-current spectrum measurement. The measured and theoretical poissonian photoelectron statistics are compared. The integrated photocurrent, A = ftt+Ti(t)dt over the measurement time interval, T, is transformed into the primary photoelectron number, n, by the relation n = A/ev~.. The theoretical poissonian distribution is calculated by P(n) = e x p ( - ( n ) ) (n)n/n !, where (n) is the average number o f primary photoelectrons and is given by (n) = lpT/e. The agreement between the experimental and theoretical poissonian statistics is fairly good. The slight deviation is due to thermal noise contribution, which was excluded in the following way. If the normalized second m o m e n t , X, of a photoelectron number is defined as X ~- [((z2~'/)2) - (n)]/(n},
(2)
then X is related to the excess intensity noise factor, X2, as
15 March 1986
X2 - 1 = X.
(3)
We measured a zero mean thermal noise electron distribution b y blocking the incident LED light and obtained the second moment o f the thermal noise electron, ((2~n)2)th . The primary photoelectron second m o m e n t , ( ( ~ ) 2 } , is obtained by subtracting the thermal noise contribution: ((ZX/'/)2} = Zn2p(n) -D2nP(n)] 2 - ((2xn)2}th .
When the photoelectron statistics, P(n), is poissonian, the value o f X is zero, which corresponds to X2 = 1. Table 1 summarizes the values o f Xl obtained from the photo-current spectrum measurement, using (1), and X2 from the photoelectron counting measurement, using (3) respectively. As can be seen, the experimental results are very close to the quantum limit of X = 1.0. The photo-current fluctuation spectra after the low pass filtering for the free-running and feedback stabilized GaAs laser output (fig. l(a)) are shown in fig. 2(a) with the calibrated quantum noise level produced by an LED light. While the photo-current fluctuation spectrum for the free-running condition is higher than the quantum noise level b y 4 - 5 dB, that for the feedback stabilization is reduced below this noise level b y 6 - 7 dB, depending on the frequency. The photoelectron statistics for the free-running and
Table 1 The excess intensity noise factors, xl, obtained from the photo-current fluctuation spectrum measurement, (1), and x2 from the photo-electron counting measurement, (3), for the three quantum noise level calibration cases using an LED light, a freerunning semiconductor laser and a feedback stabilized semiconductor laser. Detector bandwidth (Measurement time interval)
Ip
6.713 MHz (74.48 ns) 9.463 MHz (52.84 ns) 13.10 MHz (38.17 ns)
1.10 mA
Free-running GaAs laser
13.10 MHz (38.17 ns)
1.10 mA
2.747 (4.388 dB)
2.621 × 108
2.821 (4.504 dB)
Feedbackstabilized GaAs laser
13.10 MHz (38.17 ns)
1.10 mA
0.207 (-6.848 dB)
2.621 X 108
0.258 (-5.887 riB)
Calibration with a LED
"~ "3 ~9~-
1.10 mA 1.10 mA
(4)
xl (Photo-current fluctuation spectrum) 1.005 (0.02 dB) 0.9954 (-0.02 dB) 0.9908 (-0.04 dB)
{n)
x2 (Photoelectron counting)
5.114 X 108
1.0002 (0.001 dB) 1.003 (0.013 dB) 1.002 (0.008 dB)
3.628 X 108 2.621 X 108
Volume 57, number 4
OPTICS COMMUNICATIONS
15 March 1986
(a)
Free-running LD rn
0
o,1
-5
r"r-
LED
F e e d b a c k LD
-lo -15 -20
, 5
0
/ 10
15
Frequency I'MHz]
"2
(b)
v
< n > = 2.6x108
n" 4x 1()5
~°o
& n = 41104
:
°jFeedback o
o
o
o
o
2x10 5
xxx -i~Kx
-24n
xx~
xxxxx
LD
o
o
Poisson distribution
Xxxxx
-
i
-&n
0
• v.i--
&n
2An
n-
Fig. 2. (a) Normalized photo-current fluctuation spectral densities for free-running and negative feedback semiconductor laser having a single photodetector. (b) Photoelectron statistics for free-running and negative feedback semiconductor laser with a single photodetector. feedback stabilized GaAs laser output (fig. 1(a)) are compared in fig. 2(b) with the theoretical poissonian distribution. While the photoelectron statistics for the free-running condition are super-poissonian, the sub-poissonian statistics are obtained during the feedback stabilization. The excess noise factor, Xl, obtained from the photo-current fluctuation spectrum and X2, from the photoelectron statistics, are shown in table 1 for the free-running and feedback stabilization conditions. It is obvious that the observed sub-poissonian photoelectron statistics and sub-quantum limit photo-current fluctuation spectrum are much more dominant phenomena as opposed to the measurement uncertainty calibrated by using a LED light. Since the photoelectron emission process is a one-to-one correspondence with the photon arrival rate when the detector quantum efficiency is high [8], the above experi-
mental results cannot be explained without assuming that the GaAs laser output features the sub-poissonian photon statistics. The photo-current fluctuation spectra measured at terminals A and B for the free-running GaAs/ A1GaAs semiconductor laser (fig. 1(b)) are shown by the circles and plusses in fig. 3 and compared with the theoretical quantum noise level shown by a solid line. The incident waves on the two Si photodiodes exhibit smaller excess noise than the result of the single detector shown in fig. 2(a). This is because the semiconductor laser is biased higher and also because there is optical loss present between the laser and the photodiode, which contributes to the elimination of the excess noise of the incident wave. When the feedback circuit was closed, the noise spectrum at terminal A was reduced by 5 - 1 0 dB below the quantum noise level. The difference is again 293
Volume 57, number 4
OPTICS COIVlMUNICATIONS
15 March 1986
A
m
5
-~ •-~
m
.g~
without feedback
°
(C)
,~with
feedback
oooOO°
oooo~ooooooooO,, /"'N
o o o ooo
Cl 0
(B,C)
.........
o oo
aLE
*
° 2 Q ~ ~ ~ e e' '~without
~
~ 85
÷
+
+
c
o
4-
ooooooS~'~®
feedback(A,B)
-5~ with f e e d b a c k ( A )
i
+ +
_10~
0
....... 5
10
15
20
Frequency (MHz)
Fig. 3. (A) The current fluctuation spectra at terminal A. A and B correspond to the cases with and without feedback stabilization. Vuenr~isch~tS~seSledeB[~{vt2°~Z°~st~ 2~dleB 5+M+tl:/edrlmVin(B'TJmoCoU:r:;ntiflnU~t~, ~°Xnxs:PteCtr;a~! 'n~q?on~d ~ i ° ; 2 d f i Z bYst~edqu~n represent the quantum noise levels for terminals A, B and for terminal C.
much larger than the uncertainty o f less than 0.05 dB in identifying the quantum noise level. This means that the incident wave on photodiode A has its reduced noise below the standard quantum limit. On the other hand, the noise spectrum at terminal B was increased by about 3 dB above the quantum noise level. This means that the incident wave on photodiode B has its noise increased to twice that of the quantum noise level. The photoelectron counting measurement exhibits the same behavior. The photoelectron statistics measured at terminal A for free-running and feedback stabilized cases are shown by the crosses and open circles in fig. 4. The theoretical poissonian distribution is shown by a solid line. It is clear in fig. 4 that the super-poissonian photoelectron statistics of a free running laser output become sub-poissonian through feedback stabilization. The experimental photoelectron statistics give the normalized second moment X = 0.81 in a feedback stabilization case, which corresponds to the excess noise factor X = 0.19 ( 7 dB). This value is in fairly good agreement with the reduced current noise spectrum shown in fig. 3. The photon statistics measured at terminal B, on the other 294
hand, exhibit a super-poissonian distribution for feedback stabilization which is broader than that for the free-running case. Again the measured value o f X = +0.87 (or X = 1.87) is in fairly good agreement with the enhanced current noise spectrum shown in fig. 3. These experimental results can be understood as follows: If the output wave from the semiconductor laser and the zero-point fluctuation incident on a beam splitter from an open port are designated by t:1 and ~, the incidental waves on photodiodes A and B are given by
~A = (il + 0)Ix/E,
(5)
and iB = (#1 -- ()/N/fT"
(6)
Here, rl (= ro + ZX#l)exp(-i2x7)) and the zero point fluctuation 0 ( - 01 + ib2) satisfy (O1) = (02) = O.
(7)
If the photodiode quantum efficiency is unity, a d c photocurrent, iA0, is given by iA0 ----e{#Af A) = er2/2"
(8)
Volume 57, number 4
OPTICS COMMUNICATIONS
15 March 1986
0
(A)
o~ o
Xx=0.26
6x10 I
Xo = - 0 . 8 1 0 0
4x1() 5
0
%
o
2x18l i oL
-2&n
-An
0 n-
&n
2&n
i
I 8x 1()5 ~
6xl~L i
4x 1~s~ I
Xx=0.31 Xo=0.87
(B)
2x 1()5r li o
~~,~o
~ o~.,~ "~ / -2An -An
0 n-
", ~oo
~£~o An 2An
Fig. 4. The photoelectron probabilities at terminals A and B with and without feedback stabilization. The integration time interval is T = 37 ns. The solid lines are theoretical Poisson distributions for the average photoelectron number (n) = 1.5 X 108 . XXX : without feedback, ooo: with feedback. An = 2 × 104. The p h o t o c u r r e n t fluctuation, A/A, o f p h o t o d i o d e A "measures" the i n t e n s i t y f l u c t u a t i o n operator [8] as
A/B, o f the p h o t o d i o d e B "measures" A/'B = 2 e ( / B 1 ) ~ B 1 = e r 0 ( z ~ l -- Cl)-
A/A = 2 e ( i a l ) Z~A1 = ero (2g:l + Cl )-
(9)
The q u a n t u m mechanical correlation established b y feedback stabilization is
~1 = ~1-
(]o)
Consequently, the fluctuation spectra of the incident wave, i A , on p h o t o d i o d e A is reduced. On the other hand, the photo-current fluctuation,
(1 1)
Eq. (1 1) explains the e n h a n c e d i n t e n s i t y noise spectral density and super-poissonian p h o t o e l e c t r o n statistics. In order to confirm the above discussion, the photo-current f l u c t u a t i o n spectral density and photoelectron statistics were measured at terminal C, that is, the subtract o u t p u t . The dc p h o t o c u r r e n t at terminal C is expressed as 295
Volume 57, number 4
OPTICS COMMUNICATIONS
ic0 = e(<#tA#A) -- (/:B~/:B))= e(r 2 -- r02) = 0.
(12)
15 March 1986
Fmuio. Kanaya of Musashino ECL, NTT for their useful discussions.
The photocurrent fluctuation, Aic, "measures" the following operator [17] 3Jc = 2e(<#A1>2~A1 -- (#nl > ~ B 1 ) = 2er0?l •
(13)
Eq. (13) suggests that the photocurrent fluctuation, Aic, measured at terminal C reflects the beat noise between the coherent excitation of the laser output wave, r0, and the in-phase component of the zero-point fluctuation, kl" It further suggests that it is always standard quantum-limited irrespective of the feedback stabilization. This was actually confirmed by the experiment. The photo-current fluctuation spectral densities measured at terminal C for freerunning and feedback stabilization cases are shown by the crosses and triangles in fig. 3, which are just 3 dB above the q u a n t u m noise level for terminals A and B. Although the sub-poissonian state generated inside a negative feedback loop cannot be extracted by a conventional output coupling scheme as described above, extraction is possible if the q u a n t u m nondemolition (QND) measurement [18,19] is used instead of the conventional beam splitter and photodetector to measure the photon flux noise of the laser output field. The QND detector can measure the q u a n t u m state without perturbing (destroying) it. Indeed, the QND measurement of a photon number is possible via optical Kerr effect [20]. The combination of the negative feedback semiconductor laser and the QND measurement of a photon number can generate the subpoissonian state observed here outside a feedback loop [12].
References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11 | [12] [13] [14] [15] [16]
[17] [18] [19]
The authors wish to thank Prof. Hermann A. Haus of Massachusetts Institute of Technology, Prof. Olle Nilsson of the Royal Institute of Technology and Dr.
296
[20]
H. Takahashi, Adv. Commun. Syst. 1 (1965) 227. D. Stoler, Phys. Rev. D4 (1971) 1925. H.P. Yuen, Phys. Rev. A13 (1976) 2226. F. Milburn and D.F. Wails, Optics Comm. 39 (1981) 401. R.J. Glauber, Phys. Rev. 131 (1963) 2766. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277. C.M. Caves, Phys. Rev. D23 (1981) 1693. H.P. Yuen and J.H. Shapiro, IEEE Trans. Inform. Theory, IT-26 (1980) 78. R. Short and L. Mandel, Phys. Rev. Lett. 51 (1983) 384. M.C. Teich, B.E.A. Saleh and D. Stoler, Optics Comm. 46 (1983) 244; M.C. Teich and B.E.A. Saleh, J. Opt. Soc. Am. B2 (1985) 275. Y. Yamamoto, O. Nilsson and S. Saito, Proc. of 3rd USJapan Seminar on Coherence, Incoherence and Optical Chaos in Quantum Electronics, 1984, Nara, Japan, p. 7. Y. Yamamoto, N. Imoto and S. Machida, Phys. Rev. A., to be published. J. Franck and G. Hertz, Vet. Dtsch. Phys. Ges. 16 (1914)457. S. Saito, O. Nilsson and Y. Yamamoto, Appl. Phys. Lett. 46 (1985) 3. Y. Yamamoyo, O. Nilsson and S. Saito, IEEE J. Quantum Electron. QE-21 (1985) 1919. P.L. Liu, L.E. Fencil, J.S. Ko, I.P. Kaminow, T.P. Lee and C.A. Burrus, IEEE J. Quantum Electron. QE-19 (1983) 1348. H.P. Yuen and V.W.S. Chart, Optics Lett. 8 (1983) 177. V.B. Braginsky, Y.I. Vorontsov and K.S. Thorne, Science 209 (1980) 547. C.M. Caves, K.S. Thorne, R.W.P. Drever, V.D. Sandberg, and M. Zilnmermann, Rev. Mod. Phys. 52 (1980) 341. N. Imoto, H.A. Haus and Y. Yamamoto, Phys. Rev. A32 (1985) 2287.