which is non-zero, even in the case of squeezing without a coherent amplitude, where|α|=0. Note that a squeezed state with a coherent amplitude (|α| 6=0) is referred to as a bright squeezed state, while a squeezed state with no coherent amplitude is referred to as a vacuum squeezed state. The ball-and-stick representations of bright and vacuum squeezed states are shown in figure 3.3band3.3d, respectively. In the sideband picture there are correlations between the upper and lower sidebands, shown in figure3.2b.
3.3
Measuring states of light
To calculate the optical power of a state of light from the field representation, the number operator is used. The number operator is the expectation value of the Hermitian conjugate of the creation operator multiplied by the annihilation operator, N =haˆ†aˆi. The power in a field can then be calculated by multiplying the number operator by the energy of a single photon at frequencyω,
P=h¯ωhaˆ†aˆi, (3.30) where ˆa has units of pphotons/second. This section discusses methods of photodetection to determine the amplitude and phase of an electromagnetic field.
3.3.1 Photodetection
Photodection makes use of the photoelectric effect to convert photons to an electrical signal. A photon incident on photoelectric surface, such as a semiconductor, transfers energy to an electron, generating a current which can then be measured. Typically a photodiode is used as the sensor, and the current is read out using a transimpedance amplifier to convert the current to voltage. The photocurrent generated by a detector upon incidence of a beam of optical power P is given by
i=eηPD
¯
hω P (3.31)
=eηPDhaˆ†aˆi,
where e is the charge of an electron,ηPD is the quantum efficiency of the detectors, ¯hω is the
energy of a single photon at angular frequencyω.
Two modes of operation explored in this thesis are photoconductive or photovoltaic detection. In a photoconductive detector the detector is operated with a reverse bias voltage across the diode and current from the diode is measured directly. Photovoltaic detectors use a transimpedance amplifier to convert the current to voltage, giving better dark current performance.
3.3.2 Balanced homodyne detection
Balanced homodyne detection is used to measure the variances of vacuum squeezed states. Direct photodection cannot be used in the squeezed vacuum case, even though there are photons present, they are too few to generate a measurable photocurrent. The balanced homodyne technique uses a bright local oscillator which is combined optically with a signal beam on a beamsplitter, both output ports are measured on two photodiodes as illustrated in figure 3.4. Subtracting the two currents allows a measurement of the quantum noise of one input beam to be amplified by the amplitude of the other input beam. The measurement quadrature is determined by the phase difference between the two input beams,φ. This is shown mathematically below, following the treatment in [144].
Figure 3.4: General homodyne detection schematic. ˆAand ˆBare the input fields, ˆCand ˆDare the output
fields,ηBSis the reflectivity of the beamsplitter,i1andi2are the measured photocurrents from each detector.
The fields at the output ports of the beamsplitter are given by
ˆ C=√ηBSAˆ+ p 1−ηBSBeˆ iφ, Dˆ = p 1−ηBSAˆ− √ ηBSBeˆ iφ. (3.32)
where ηBS is the reflectivity of the beamsplitter, and ˆA and ˆB are the input fields. The number
operators of the two output fields are given by
ˆ C†Cˆ=ηBSAˆ†Aˆ+ p ηBS(1−ηBS)(Aˆ†Beˆ −iφ+Bˆ†Aeˆ iφ) + (1−ηBS)Bˆ†B,ˆ (3.33) ˆ D†Dˆ = (1−ηBS)Aˆ†Aˆ− p ηBS(1−ηBS)(Aˆ†Beˆ iφ+Bˆ†Aeˆ −iφ) +ηBSBˆ†B.ˆ (3.34)
The photocurrents generated by each diode are proportional to the number operator of the field incident on the diode, that is i1 ∝Cˆ†Cˆ and i2 ∝Dˆ†Dˆ. The photocurrents are subtracted
electronically. Experimentally either the subtracted current is read out directly, or the two currents are measured separately and then subtracted. The subtracted current is given by
i1−i2= (2ηBS−1)Aˆ†Aˆ+ (1−2ηBS)Bˆ†Bˆ+2
p
ηBS(1−ηBS)(Aˆ†Beˆ iφ+Bˆ†Aeˆ −iφ). (3.35)
The fields may be linearised by separating them into their steady state and time-varying com- ponents. Assuming that the time varying component of the field is much smaller than the steady state field, second order fluctuating terms can be neglected. The linearised input fields are given by ˆA=a¯+δaand ˆB=b¯+δb, thus ˆ A†Aˆ= (a¯†+δa†)(a¯+δa) (3.36) =|a¯|2+a¯†δa+a¯δa† =|a¯|2+a δXˆA+,
recallingδXˆA+= (δAˆ+δAˆ†). The ˆBfield can be treated similarly, hence the subtracted photocur- rent may be rewritten in terms of quadrature operators,
i1−i2= (2ηBS−1)(|a¯|2+aδXˆA+) + (1−2ηBS)(|b¯|2+bδXˆB+) (3.37)
+2pηBS(1−ηBS)(2abcosφ+a¯(δXˆB+cosφ−δXˆ
−
B sinφ)
3.3 Measuring states of light
Assuming a perfect beamsplitter, withηBS =0.5, eliminates the first two terms of equation
3.37. The remaining terms represent the DC field proportional to the phase difference between the two beams. The cross terms imply that the noise in ˆAis amplified by the magnitude ¯band the noise in ˆBis amplified by the magnitude of ¯a. In the case of homodyne detection of vacuum squeezed states, such as in this thesis, we can also assume that ¯ba¯, and hence the subtracted current is given by
i1−i2'2 ¯ab¯cosφ+b¯(δXˆA+cosφ−δXˆA−sinφ). (3.38)
Note that the phase difference between the two input fields determines which quadrature of ˆA
is being observed.
3.3.3 Power spectral density
To extract frequency domain information of a signal the power spectral density of the noise is calculated. Experimentally this is done either in post-processing of a time-domain signal or using a spectrum analyser. The single-sided noise spectral density is defined as the Fourier transform of the time-domain auto-correlation function, and gives a measure of the noise of an operator as a function of Fourier frequency. The continuous Fourier transform of a time-domain operator ˆa(t) is given by ˜ a(ω) = Z ∞ −∞ ˆ a(t)eiωtdt, (3.39)
whereω is the Fourier frequency, relative to the carrier frequency. The power spectral density,
S(ω), is then S(ω) = 1 2π Z τ G(τ)eiω τd τ, (3.40)
whereG(τ) =hδa(t)δa(t+τ)iis the autocorrelation between the time-varying component of ˆa(t) over the measurement time, τ. Note that δa(t) is only the time-varying component of ˆa(t), and has an average value of zero.
Measurements of the power spectral density are generally taken over some finite resolution bandwidth B about the carrier frequency ω. B is chosen so that the power spectral density is roughly constant over this frequency band. It can be shown [26, 169] that the power spectral density over this bandwidth is given by
SB(ω) =S(ω)×B. (3.41)
Thus such measurements must be normalised to the measurement bandwidth, the normalised power spectrum is given by
V(ω) =SB(ω)
B =h|δX˜i(ω)|
2i, (3.42)
whereδX˜i(ω)is the fluctuating component of the frequency-domain quadrature operator ˜Xof the
ith quadrature. The power spectral densityV(ω)is equivalent to the variance of a time-domain signal, thus for vacuum statesV(ω) =1.
Figure 3.5: Phasor diagram illustrating phase noise. ∆Xˆ−is the squeezed quadrature, ∆Xˆ+ is the anti-
squeezed quadrature,∆Xˆθand∆Xˆθ+π/2are two orthogonal quadratures. Jitter on the phase of the squeezing
ellipse causes coupling between the phase and amplitude quadratures.
3.3.4 Loss
Loss occurs when any light is coupled out of the main mode of a beam, by absorption, scattering, or imperfect reflection and transmission of optical components. Sources of loss can be modelled as a beamsplitter with non-unity reflectivity, and one input port consisting of vacuum, as in figure 3.4, letting ˆBfield be a vacuum state with fluctuating componentδ ν. Assuming the beamsplitter is highly reflective, only the ˆCfield leaving the beamsplitter need be considered.
ˆ
C=√ηloss(a¯+δa) +
p
1−ηlossδ νeiφ. (3.43)
When a squeezed state encounters a source of loss, vacuum is coupled to the field, and the state is no longer minimum uncertainty. In terms of variances of a field, the effect of loss is given by
Vtot=ηlossVin+ (1−ηloss)Vvac. (3.44)
WhereVtotis the output variance after accounting for the loss,Vinis the initial variance without
loss,ηlossis the detection efficiency,Vvacis variance of the vacuum, which is unity.
3.3.5 Phase noise
Phase noise, also known as squeezed-quadrature fluctuations or squeezing angle jitter, refers to fluctuations in the angle of the squeezing ellipse. When quadrature fluctuations occur faster than the measurement time, the anti-squeezed quadrature couples to the squeezed quadrature, and the measured level of squeezing is reduced, as shown in figure3.5.
In terms of quadrature variances (defined in equation 3.13with respect to the uncertainty in an arbitrary quadrature) the effect of a level of RMS phase noise ˜θsqz on an arbitrary quadrature
V(θsqz)is given by
Vtot(θsqz,θ˜sqz) =V(θsqz)cos2θ˜sqz+V(θsqz+π/2)sin2θ˜sqz, (3.45)
whereV(θsqz)andV(θsqz+π/2)are the variances in the cosine and sine quadratures in the absence of phase noise.