Versatile Wideband Balanced Detector for Quantum Optical Homodyne Tomography

Abstract

We present a comprehensive theory and an easy to follow method for the design
and construction of a wideband homodyne detector for time-domain quantum
measurements. We show how one can evaluate the performance of a detector in a
specific time-domain experiment based on electronic spectral characteristic of
that detector. We then present and characterize a high-performance detector
constructed using inexpensive, commercially available components such as
low-noise high-speed operational amplifiers and high-bandwidth photodiodes. Our
detector shows linear behavior up to a level of over 13 dB clearance between
shot noise and electronic noise, in the range from DC to 100 MHz. The detector
can be used for measuring quantum optical field quadratures both in the
continuous-wave and pulsed regimes with pulse repetition rates up to about 250
MHz.

Full text

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arXiv:1111.4012v1 [physics.ins-det] 17 Nov 2011
Versatile Wideband Balanced Detector for Quantum Optical Homodyne Tomography
R. Kumar, E. Barrios, A. MacRae, and A. I. Lvovsky
Institute for Quantum Information Science, University of Calgary, Calgary, Canada, T2N 1N4
E. Cairns
Department of Chemistry, University of Calgary, Calgary, Canada, T2N 1N4
E. H. Huntington
School of Engineering and Information Technology, University College,
The University of New South Wales, Canberra ACT 2600, Australia
(Dated: November 18, 2011)
We present a comprehensive theory and an easy to follow method for the design and construction
of a wideband homodyne detector for time-domain quantum measurements. We show how one can
evaluate the performance of a detector in a specific time-domain experiment based on electronic
spectral characteristic of that detector. We then present and characterize a high-performance detec-
tor constructed using inexpensive, commercially available components such as low-noise high-speed
operational amplifiers and high-bandwidth photodiodes. Our detector shows linear behavior up to
a level of over 13 dB clearance between shot noise and electronic noise, in the range from DC to
100 MHz. The detector can be used for measuring quantum optical field quadratures both in the
continuous-wave and pulsed regimes with pulse repetition rates up to about 250 MHz.
I.INTRODUCTION
The balanced homodyne detector (HD) is a useful tool
in quantum optics and quantum information processing
with continuous variables [1, 2] since it can be used to
measure field quadratures of an electromagnetic mode.
These measurements provide information for complete
reconstruction of quantum states in the optical domain
(optical homodyne tomography).
With developing tools of continuous-wave quantum-
optical state engineering [3] as well as state and process
tomography [4], the performance requirements for homo-
dyne detectors continue to increase. The design of HDs
for time-domain quantum tomography [5–7] is based on
four main performance criteria: a) high bandwidth and a
flat amplification profile within that bandwidth; b) high
ratio of the measured quantum noise over the electronic
noise; c) high common mode rejection ratio (CMRR); d)
quantum efficiency of the photodiodes.
The high bandwidth requirement comes from the fact
that an HD must be able to measure field quadratures
with sufficient time resolution.
lasers, this corresponds to the inverse of the repetition
rate of the pulses; in the case of a continuous signal, the
required resolution is determined by the duration of the
optical mode in which the signal states are produced [8–
10]. This is technically challenging because most ampli-
fiers have a limited gain-bandwidth product. Increasing
the bandwidth implies reducing the gain, which, in turn,
increases the effect of the electronic noise. Also the high
frequency circuit layout poses a challenge to designers.
Within its bandwidth range, the HD must feature a flat
amplification profile. If this is not the case, the response
of the HD to each individual pulse will exhibit ringing,
which degrade the detector’s time resolution and distort
the measurement. This requirement also presents a ma-
In the case of pulsed
jor design challenge.
Any non-desirable ambient noises, dark current noises
from the photodiodes and the intrinsic noise of the am-
plifiers fall under the umbrella of electronic noise. The
effect of this noise is to add a random quantity Qeto the
measurement of the field quadrature Qmeas. This effect
is equivalent to an additional optical loss channel with
transmission [11]
ηe= 1 − ?ˆQ2
e?/?ˆQ2
meas?. (1)
As we show below, the value of ηedepends not only on the
characteristics of the detector, but also on the conditions
of the measurement in which the detector is used.
A HD must have a high subtraction capability be-
tween the two photocurrents produced by the photodi-
odes. This can be expressed as a generalized common
mode rejection ratio (CMRR) of the balanced detection
[6, 12]. The CMRR measures the ability of the device
to reject the classical noise of the local oscillator [8, 13].
This is particularly important in the pulsed case because
a low CMRR (which implies a poor subtraction) will re-
sult in contamination of the signal with the repetition
rate of the pulse and harmonics. Additionally, this lack
of subtraction capability will make the HD more sus-
ceptible to saturation by the amplified signal from the
photodiodes. High CMRR is difficult to achieve because
the response functions of the photodiodes are not exactly
the same. Therefore a pair of photodiodes with response
functions as similar as possible must be chosen.
Experimentally, these performance benchmarks can be
measured using an electronic spectrum analyzer. The
spectrum of the homodyne output photocurrent gives in-
formation about the detector’s bandwidth and amplifi-
cation profile. Observing the output current in the ab-
sence of the local oscillator provides information about
the magnitude and spectrum of the electronic noise. The

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lower bound on CMRR is determined by comparing the
HD spectra when both photodiodes are illuminated and
when only one is illuminated while the other is blocked.
In the present work, we quantitatively relate the mea-
sured electronic spectra to added noise in quadrature
measurements. We show that the limited bandwidth and
electronic noise can be translated into equivalent optical
losses such as in Eq. (1). We show how to estimate and re-
duce these losses for a specific time-domain experiment.
In fact, in many cases (particularly, in the continuous-
wave regime) electronic spectral measurements on the
HD photocurrent in the presence and absence of the local
oscillator are sufficient to precisely calculate the equiva-
lent loss associated with the electronics.
The theoretical discussion in this paper is limited to
the effects of the bandwidth and the electronic noise in
two practically relevant regimes. The effect of the non-
unitary quantum efficiency on quantum state reconstruc-
tion is well known [2]. A discussion of CMRR has been
presented in detail in Ref. [12].
We then demonstrate an easy to follow method for
the design and construction of a wideband homodyne
detector using commercial available components such
as low-noise high-speed operational amplifiers and high-
bandwidth photodiodes. Aside from high performance
benchmarks, a special feature of our detector is its ver-
satility: it is designed and tested to operate in both the
continuous-wave or pulsed regimes. Therefore the unit
presented here may be useful for a wide range of quan-
tum optics experiments.
II.THEORETICAL ANALYSIS
Balanced homodyne detection consists of overlapping
the signal mode carrying the quantum state in question
and a strong reference field in a matching mode (the lo-
cal oscillator, or LO) on a symmetric beam splitter. The
two output signals of this beam splitter are directed to
the two photodiodes of the HD, where these fields are
detected and subtracted. Neglecting experimental im-
perfections, the subtraction photocurrent is then
ˆi(t) = Aα(t)ˆ qθ(t), (2)
where ˆ qθ(t) is the instantaneous field quadrature value in
the signal mode, α(t) and θ are the local oscillator ampli-
tude and phase, respectively, and A is a proportionality
coefficient related to the HD amplifier gain. It is assumed
that the local oscillator phase is constant.
The instantaneous quadrature observable can be writ-
ten as
ˆ qθ(t) = ˆ a(t)eiθ+ ˆ a†(t)e−iθ, (3)
where ˆ a(t) is the time dependent photon annihilation
operator[14].
In a practical HD, the relationship between the quadra-
ture measurement and the output current is more com-
plex. It can be approximated by
ˆi(t) =ˆie(t) + A
+∞
?
−∞
α(t′)ˆ q(t′)r(t − t′)dt′, (4)
where ie(t) is the detector’s electronic noise and r(·) is its
response function. An ideal detector would have ie(t) = 0
and r(τ) = δ(τ). In practice these conditions are not met.
As evident from Eq. (4), the impact of the electronic
noise is minimized by raising the power of the local os-
cillator and the amplifier gain. However, practical possi-
bilities of increasing the gain without proportionally in-
creasing the electronic noise are limited. The local oscil-
lator power must also be restricted to avoid saturation of
the photodiodes and eliminating the classical noise [13].
Therefore in the analysis below we assume A and α to
equal their optimal values for the given experimental set-
ting.
A.Continuous regime
In the continuous regime, the amplitude of the local
oscillator is a constant: α(t) ≡ α. We are interested in
measuring the quadrature of the signal field associated
with a particular (normalized) temporal mode function
φ(t), which we assume real:
ˆQ =
+∞
?
−∞
ˆ q(t)φ(t)dt (5)
To that end, we integrate the homodyne photocurrent
with a certain weight function ψ(t), obtaining a measured
quadrature value,
ˆQmeas=
+∞
?
−∞
ˆi(t)ψ(t)dt (6)
= Aα
+∞
?
−∞
+∞
?
−∞
ˆ q(t′)ψ(t)r(t − t′)dtdt′+ˆQe,
in which the last term,
ˆQe=
+∞
?
−∞
ˆie(t)ψ(t)dt,(7)
corresponds to the electronic noise contribution, which
we will discuss later. First, we discuss the effect of finite
detector response function (bandwidth) on the quadra-
ture measurement.
Equation (6) can be rewritten as
ˆQmeas= Aα
+∞
?
−∞
ˆ q(t′)ψ′(t′)dt′+ˆQe, (8)

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where
ψ′(t′) =
+∞
?
−∞
ψ(t)r(t − t′)dt. (9)
By comparing Eqs. (5) and (8) we find that, by choosing
ψ(t) such that ψ′(t) = φ(t), we haveˆQmeas= AαˆQ+ˆQe,
i.e., the distortions associated with the detector’s finite
bandwidth are completely eliminated. This may however
be difficult in practice, because the required weight func-
tion is a deconvolution of the temporal mode of interest
and the detector’s response. Lack of precise knowledge
of either of the above may lead to significant errors in
deconvolving.
If ψ′(t) ?= φ(t), the detection efficiency is degraded by
the mode matching factor [15]
ηb=
??????
+∞
?
−∞
ψ′(t)φ(t)dt
??????
2?+∞
−∞
?
|ψ′(t)|2dt ,(10)
where the denominator normalizes ψ′(t). A practically
important particular case is when the temporal mode of
the signal is known and the finite bandwidth of the de-
tector is neglected, so ψ(t) is set to equal φ(t). In Fig. 1,
the efficiency (10) obtained in this setting is plotted for
Gaussian φ(t) and r(t) as a function of the detector band-
width, which, as we show below, is obtained from the
Fourier transform of the response function. As we see,
the detector bandwidth has no significant degrading ef-
fect on the measurement (ηb> 0.99) as long as it is com-
parable to or larger than the inverse temporal width of
the signal temporal mode.
Bandwidth?(in?units?of?inverse?FWHM
of?the?signal?temporal?mode)
Effective?efficiency ?b
0.20.40.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
FIG. 1. Effective efficiency (10) of the HD associated with
its finite bandwidth. The 3-dB bandwidth is plotted along
the horizontal axis in units of the inverse full-width-at-half-
maximum of the signal temporal mode φ(t). Both functions
are assumed Gaussian.
Let us now calculate the contribution of the electronic
noise to the measured quadrature, so the equivalent ef-
ficiency (1) can be estimated. We start with the elec-
tronic spectra of the HD output photocurrent in the ab-
sence and in the presence of the local oscillator, with the
signal in the vacuum state. According to the Wiener-
Khintchine theorem, these spectra are given, respectively,
by
Se(ν) =
+∞
?
−∞
?ˆie(t)ˆie(t + τ)?e2πiντdτ(11)
and
S(ν) =
+∞
?
−∞
?ˆi(t)ˆi(t + τ)?e2πiντdτ, (12)
where ν is the frequency and the averaging is performed
over both time t and the quantum ensemble of the vac-
uum signal state. The autocorrelation function in the
latter equation can be further simplified as
?ˆi(t)ˆi(t + τ)? (13)
= A2α2
?+∞
−∞
?
+∞
?
−∞
ˆ q(t′)r(t − t′)ˆ q(t′′)r(t + τ − t′′)dt′dt′′
?
+?ˆie(t)ˆie(t + τ)?
= A2α2
+∞
?
−∞
?r(t − t′)r(t + τ − t′)?dt′+ ?ˆie(t)ˆie(t + τ)?
= A2α2
+∞
?
−∞
r(t)r(t + τ)dt + ?ˆie(t)ˆie(t + τ)?
because, in the vacuum state, ?ˆ q(t′)ˆ q(t′′)? = δ(t′− t′′).
From the above, we find
S(ν) = A2α2|˜ r(ν)|2+ Se(ν),(14)
with
˜ r(ν) =
+∞
?
−∞
r(t)e2πiντdt(15)
being the Fourier image of r(t). In other words, neglect-
ing the electronic noise, the spectrum of the HD output
current in the continuous regime is simply the squared
amplitude of the Fourier transform of the detector’s re-
sponse function. Note, however, that the response func-
tion cannot be obtained from this spectrum because in-
verse Fourier transformation also requires data on the
phase of ˜ r(ν). The response function can be measured
directly in the time domain with a pulsed LO as discussed
below.
Let us now discuss the contribution of electronic noise
to quadrature measurements. From Eq. (8), and because

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in the vacuum state ?ˆ q(t)ˆ q(t′)? = δ(t − t′), we can write
?ˆQ2
meas? = A2α2
+∞
?
−∞
|ψ′(t)|2dt + ?ˆQ2
e? (16)
= A2α2
+∞
?
−∞
|˜ψ′(ν)|2dν + ?ˆQ2
e?
= A2α2
+∞
?
−∞
|˜ψ(ν)|2|˜ r(ν)|2dν + ?ˆQ2
e?,
where the variance of the electronic noise component is
given by
?ˆQ2
e? =
+∞
?
−∞
+∞
?
−∞
?ˆie(t)ˆie(t + τ)?ψ(t)ψ(t + τ)dtdτ
=
+∞
?
−∞
Se(ν)|˜ψ(ν)|2dν.(17)
Combining the above two results with Eq. (14), we find
?ˆQ2
meas? =
+∞
?
−∞
S(ν)|˜ψ(ν)|2dν. (18)
Equations (17) and (18) lead us to an important con-
clusion: by knowing the homodyne output spectra S(ν)
and Se(ν), as well as the weight function ψ(t), one
can predict the fraction of the electronic noise in the
measured quadrature variance in an arbitrary temporal
mode. This, as discussed above, directly translates into
an equivalent optical loss.
In the case of a high-bandwidth detector, when S(ν)
and Se(ν) can be assumed constant over the support of
˜ψ(ν), we have
1 − ηe≈
?Se(ν)
S(ν)
?
signal bandwidth
.(19)
This quantity, which we call clearance of the detector’s
shot noise over the electronic noise, is one of the primary
characteristics of any HD circuit.
B.Pulsed regime
Now let us suppose that the LO is pulsed, with the
pulse width much shorter than the time resolution of the
electronics. In this case, Eq. (4) takes the form
ˆi(t) = AαpˆQr(t) +ˆie(t),(20)
where the pulse is assumed to occur at t = 0, ˆQ =
+∞
?
−∞
α(t)ˆ q(t)dt/αpis the normalized quadrature operator
corresponding to the signal mode defined by the shape
of the LO pulse, with αp=
?
+∞
?
−∞
|α(t)|2dt being the ef-
fective amplitude of the local oscillator pulse. In other
words, neglecting the electronic noise, the shape of the
HD response to a single short pulse is given by the de-
tector’s response function.
The quadrature measurement is obtained by integrat-
ing the homodyne photocurrent over a certain time in-
terval:
ˆQmeas=
t2
?
t1
ˆi(t)dt = AαpˆQ
t2
?
t1
r(t)dt +ˆQe,(21)
with
ˆQe=
t2
?
t1
ˆie(t)dt.
The optimal choice of the integration limits is determined
by the bandwidths of the detector’s electronic noise and
the temporal width of its response function.
When the local oscillator is a train of pulses with rep-
etition period T, the HD output current is given by
ˆi(t) = Aαp
∞
?
j=−∞
ˆQjr(t − jT) +ˆie(t),(22)
whereˆQjis the quadrature operator of the jth pulsed sig-
nal mode, with the pulse of interest having index j = 0.
If the response function is nonzero over an interval longer
than T, the quadrature measurement is contaminated by
that of the neighboring pulses:
ˆQmeas,0= Aαp
∞
?
j=−∞
RjˆQj+ˆQe,(23)
where Rj=
t2 ?
t1
r(t−jT)dt. The sum in Eq. (23) defines a
new measured mode whose state is not necessarily identi-
cal to that in the j = 0th pulsed mode. The correspond-
ing mode matching efficiency (neglecting the electronic
noise) is given by
ηb=
R2
∞
?
0
j=−∞R2
j
.(24)
This efficiency is plotted in Fig. 2 for the response func-
tion of Gaussian shape as a function of the 3-dB band-
width of the detector response function spectrum. As we
see, a detector bandwidth of at least 0.4/T is required
for ηbto exceed 99%.
If r(t) is known, so are all Rj and the effect of finite
bandwidth can be reversed by means of discrete decon-
volution, akin to the continuous case. However, partial

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0.1 0.20.30.40.5
0.5
0.6
0.7
0.8
0.9
1.0
Bandwidth?(in?units?of?repetition?rate)
Effective?efficiency ?b
FIG. 2. Effective efficiency of the HD associated with tempo-
ral overlap of responses to different pulsed modes, as a func-
tion of the 3-dB bandwidth of the electronics. The response
function is assumed Gaussian. The solid line corresponds to
a short integration interval [−ǫ,+ǫ]; the dashed line to the
integration interval of length T. Both integration intervals
are centered at the peak of the response function.
reversal can be implemented even if the response func-
tion is not known, provided that ηb is sufficiently high,
i.e. |Rj| ≪ |R0| for j ?= 0, as follows. In a typical de-
tector, Rj are negligibly small for j > 0: most of the
ringings occur after the optical pulse that generates the
response. This is the situation, for example, with the
detector assembled in this work.Then we have, according
to Eq. (23), and because ?ˆQjˆQk? = δjk in the vacuum
state,
?ˆQmeas,0ˆQmeas,i? = A2α2
p
?
j
RjRj−i≈ A2α2
pR0R−i,
(25)
whereˆQmeas,i denotes the quadrature measurement for
the ith pulse. The above correlation can be easily ob-
tained experimentally, from which one can determine
R−i
R0
≈?ˆQmeas,0ˆQmeas,i?
?ˆQmeas,0ˆQmeas,0?. (26)
One then calculates
Q′
meas,0= Qmeas,0−
∞
?
i=1
?ˆQmeas,0ˆQmeas,i?
?ˆQmeas,0ˆQmeas,0?Qmeas,i
(27)
for each experimentally measured quadrature, thereby
eliminating contamination from neighboring pulses. The
resulting quadrature values are then renormalized and
used in quantum state reconstruction.
this technique, the reconstruction efficiency has been im-
proved in Ref. [16].
We now use Eq. (12) to determine the spectral power
of the HD output in the pulsed regime. The ensemble
average of i(t)i(t + τ) is a periodic function of time t,
By means of
hence we can write
?ˆi(t)ˆi(t + τ)? =1
T(Aαp)2
(28)
×
∞
?
j,k=−∞
?ˆQjˆQk?
T/2
?
−T/2
r(t − jT)r(t + τ − kT)dt
+?ˆie(t)ˆie(t + τ)?t.
In the vacuum state,
?ˆi(t)ˆi(t+τ)? =1
T(Aαp)2
+∞
?
−∞
r(t)r(t+τ)dt+?ˆie(t)ˆie(t+τ)?t.
(29)
Accordingly,
S(ν) =1
T(Aαp)2|˜ r(ν)|2+ Se(ν).(30)
We see that in spite of the pulsed character of the local
oscillator, the HD spectrum is determined by the Fourier
transform of its response function akin to the continuous
case. An important difference is the multiplication by
the pulse repetition rate: when the separation between
the pulses is increased, the spectral power reduces pro-
portionally.
In contrast to the continuous regime, in the pulsed
case the evaluation of the equivalent efficiency (1) re-
quires knowledge of r(t); information on the spectra S(ν)
and Se(ν) is not sufficient. Let us, however, consider
a practically important particular case when the band-
width of both the detector’s response and the electronic
noise greatly exceed the laser repetition rate. Suppose
the integration in Eq. (21) is done over the time interval
[−T0/2,T0/2] with T0< T. Then we have
?ˆQ2
meas? =
?

AαpˆQ
T0/2
?
−T0/2
r(t)dt



2?
+ ?ˆQ2
e?. (31)
We assume that the temporal width of the function r(t)
is much less than T0, so the integration limits can be
replaced by ±∞. We then have
?ˆQ2
meas? = (Aαp)2|˜ r(0)|2+ ?ˆQ2
e?, (32)