### Appendix

### 1.1 A.1 Locking

To preform measurements that are able to saturate the QCRB, we need to stabilize multiple aspects of the experiment such as the number of photons used to probe the system under study and the transmission modulation amplitude. To keep the number of probing photons constant, the power of the probe beam after the FWM process needs to be stabilized. This requires keeping both the seed probe power and the gain of the FWM processes stable.

To stabilize the power of the seed probe for the FWM process, a portion of the probe beam is picked off via a half waveplate and PBS before the Rb vapor cell. This pick off is then detected with a photodiode and is used to obtain the error signal. The power of the seed probe is kept constant by controlling the diffraction efficiency of the AOM used to generate the probe from the pump beam. This setup is typically referred to as a noise-eater.

Given that the gain of the FWM process depends on the atomic number density, pump power, and frequencies of the involved fields, all of these have to be stabilized. The atomic number density depends on the temperature of the Rb vapor cell. Thus, a temperature controller is used to stabilize the Rb cell temperature to 120°C within a fraction of a degree. As with the probe beam, the pump power is locked before the Rb vapor cell by detecting a small portion of the pump that is picked off via a beam sampler. The pump power is then kept constant by feeding back to an electronically controlled rotation mount containing a half waveplate placed before a PBS. This feedback control is slow; however, the pump power changes are mostly due to slower thermal drifts and the intensity noise of the pump has little effect on the generated quantum state. Thus, a high bandwidth noise-eater like the one used for the probe, is not needed. Finally, since a change in the frequency of the laser changes the gain and thus the output conjugate and probe powers, we use the conjugate power to obtain the error signal to compensate for the frequency drifts. We use this approach as the probe transmission, and thus detected power, is changed as part of the experiment. The error signal obtained from the conjugate power is then fed back to lock the laser frequency, which results in a stable lock over the more than 20 hours needed to take the data.

The transmission modulation amplitude must also be controlled for reproducibility from data set to data set. This requires tight control of the transmission modulation amplitude ramp and control over the mean transmission through the EOM. In the implementation of the system, as shown in the “system” inset of Fig. 1, the EOM setup has a quarter waveplate and a half waveplate before the EOM and a half waveplate and a PBS after the EOM. The waveplates before the EOM, used for the amplitude modulation configuration, are set such that the EOM has a high transmission and the transmission modulation amplitude of the system, *δT*, is within the linear regime, as can be seen in Fig. 4. As such, the maximum, \(T_{\max }\), and minimum, \(T_{\min }\), transmissions through the EOM setup have to be properly set. For our experiment the ratio \(T_{\max }/T_{\min }\) is set to around 1.02 with a mean transmission of 84%, giving a maximum change in transmission through the EOM of \(\approx 0.8\%\) via the applied voltage. The maximum mean transmission through the system is limited by losses introduced by the various optical elements, in particular the EOM and the two PBSs shown in the “system” inset in Fig. 1. To lock the EOM, we detect the output reflection of the PBS of the amplitude modulation configuration. The output of the photodetector is split into DC and AC signals to lock the mean transmission and transmission modulation amplitude, respectively. The DC lock ensures the EOM operates in the linear regime by keeping the mean transmission at its center value, see Fig. 4. Locking the mean transmission at the center point, *T̄* in Fig. 4, provides the largest possible slope to maximize the transmission modulation amplitude for a given modulation voltage. Furthermore, if the mean transmission drifts, the calibration of the transmission modulation amplitude (see Appendix A.2) is no longer valid. The AC lock is used to control the transmission modulation amplitude during the amplitude ramp. The AC lock PID output is sent to the function generator used to create the modulation voltage sent to the EOM. The AC and DC voltage outputs from the locking electronics are combined with a Bias Tee. The combined signal is then amplified and sent to the EOM.

Another important aspect of being able to reach the QCRB is to filter out the classical technical noise in the optical state used to probe the system under study. While the power of the seed probe beam was stabilized with a noise eater, such a configuration is unable to reduce the intensity noise to the shot noise limit [55]. As a result, a cleanup cavity (Newport SuperCavity model SR-140-C) is used to reach the shot noise limit at our operating frequency of 1.5 MHz. We use the Pound-Drever-Hall [56, 57] locking technique to keep the cleanup cavity on resonance with the probe. An EOM is used to add a phase modulation to the probe mode before coupling it into the cleanup cavity. The modulation frequency is set to 10 MHz, which is significantly larger than the linewidth of the cavity (<0.6 MHz). As a result, the sidebands from the modulation are not transmitted by the cavity and the reflected light can be used to generate the required error signal.

### 1.2 A.2 Calibrations

In order to compare the measured transmission uncertainties with the theoretical QCRB without any free parameters, the photon flux, propagation transmissions, and transmission modulation amplitude need to be properly calibrated.

To measure the propagation transmission we use two power meters. The first power meter, PM-A, is set on a flip mount in front of the Rb cell while the other, PM-B, is used to measure the transmission. First, we perform a relative calibration of the two power meters. We do this by placing PM-B right behind PM-A and measuring the power multiple times with both power meters by flipping PM-A in and out of the beam path. The ratio of the measured powers are then used to remove any systematic bias in the measurements. We then measure the transmission of the probe mode before the system under study without the Rb cell in place to be \(T_{\text{common}}=98.4\%\pm 1\%\). This path has many common optical elements for the probe and conjugate mode paths. We then place the Rb cell back in place and, working off resonance, find the transmission for each of the cell windows to be \(T_{\text{window}}=98.8\%\pm 1\%\). The quantum efficiency of the photodiodes are taken from previous calibration results to be \(\eta =94.5\%\pm 2\%\) [41]. Altogether this leads to a probe transmission before the system of \(T_{p}=T_{\text{common}}T_{\text{window}}=97.3\%\pm 1\%\), probe transmission after the system of \(\eta _{p}=\eta =94.5\%\pm 2\%\), and a conjugate transmission of \(\eta _{c}=T_{\text{common}}T_{\text{window}}\eta =91.9\%\pm 2\%\).

We operate the EOM in a regime in which the transmission modulation it introduces (\(\delta _{T}\)) is linear with the voltage applied across the EOM crystal. To calibrate the slope and thus the relation between \(\delta _{T}\) and the applied voltage, we record the probe power oscillations on an oscilloscope for a given set of 14 modulation amplitude lockpoints. We then perform a fast Fourier transform of the recorded traces to isolate the amplitude of the oscillation from the noise and fit the transmission modulation to the applied voltage oscillation amplitude, as shown in Fig. 5. We consider a maximum modulation lockpoint of 300%, corresponding to a value three times higher than the maximum lockpoint value used in the experiment, due to the high electronic noise of the oscilloscope as compared to the spectrum analyzer. As can be seen in Fig. 5, the transmission modulation is linear with the applied voltage even at the higher lockpoints used for the calibration.

Finally, the transmission modulation amplitude due to the whole system under study, *δT*, for a given mean transmission, *T*, is given by \(\delta T=T\delta _{T}\). As the mean transmission is reduced during each data set, the transmission modulation also decreases. This leads to the difference in scale between the *x*-axis of Fig. 2 and the *y*-axis of Fig. 5 in the range between 0% and 100% of the modulation lockpoint.

### 1.3 A.3 Estimation of transmission uncertainty through modulation measurements

Given that we are interested in determining the uncertainty in the estimation of transmission, we now outline the procedure to obtain its value from the transmission modulation amplitude introduced by the system under study at a SNR = 1. To do this, we consider a small transmission modulation amplitude *δT* (defined as the standard deviation of the modulation) introduced by the system under study, which results in the modulation of the measured quantity *M* according to

$$ \delta M=\delta T \biggl\vert \frac{\partial M}{\partial T} \biggr\vert , $$

(14)

where *δM* is the blue trace in Fig. 2 and \(\vert \partial M/\partial T \vert \) is the slope of that trace. Looking specifically at our measurement technique, we have that

$$\begin{aligned}& M = T \langle \hat{n}_{p} \rangle -g \langle \hat{n}_{c} \rangle , \end{aligned}$$

(15)

$$\begin{aligned}& \biggl\vert \frac{\partial M}{\partial T} \biggr\vert = \langle \hat{n}_{p} \rangle , \end{aligned}$$

(16)

where *g* in Eq. (15) is some electronic gain, and

$$ \bigl\langle \Delta ^{2}M \bigr\rangle =T^{2} \bigl\langle \Delta ^{2} \hat{n}_{p} \bigr\rangle +T(1-T) \langle \hat{n}_{p} \rangle - \frac{T^{2}\text{Cov}^{2}}{ \langle \Delta ^{2}\hat{n}_{c} \rangle } $$

(17)

for the optimal gain, with the term Cov given by the covariance between \(\hat{n}_{p}\) and \(\hat{n}_{c}\). For the bTMSS, it can be shown that

$$\begin{aligned} \frac{\delta M^{2}}{ \langle \Delta ^{2}M \rangle } =& \frac{\delta T^{2} \vert \alpha \vert ^{2}\cosh ^{2}(s)}{T^{2}\cosh (2s)+T(1-T)-T^{2} \left[\cfrac{\sinh ^{2}(2s)}{\cosh (2s)} \right]} \end{aligned}$$

(18)

$$\begin{aligned} =& \frac{\delta T^{2} \langle \hat{n}_{p} \rangle }{T-T^{2} [1-\operatorname{sech}(2s) ]} \end{aligned}$$

(19)

$$\begin{aligned} =&\frac{\delta T^{2}}{ \langle \Delta ^{2}T \rangle }, \end{aligned}$$

(20)

where the last equality is obtained by using the uncertainty in the estimation of transmission given by the QCRB for the bTMSS, \(\langle \Delta ^{2}T \rangle \), as we have previously shown in [35]. Thus, the SNR of the measurement is equal to the SNR for transmission estimation.

It is important to note that such an equality is not specific to our measurement strategy and can be shown to be the case in general through error propagation; that is

$$ \text{SNR}=\frac{\delta M}{\Delta M}= \frac{\delta T \left \vert \frac{\partial M}{\partial T} \right \vert }{\Delta M}= \frac{\delta T}{\Delta T}. $$

(21)

Thus, when SNR = 1, the transmission modulation amplitude introduced by the system is equal to the standard deviation in the estimation of transmission, such that

$$ \Delta T=\delta T|_{\text{SNR}=1}. $$

(22)

We can further relate Δ*T* to the measured signal by combining Eqs. (22) and (21), such that when SNR = 1 we have that

$$ \Delta T=\Delta M \Big/ \biggl\vert \frac{\partial M}{\partial T} \biggr\vert , $$

(23)

with the uncertainty in the estimation of transmission then given by \(\langle \Delta ^{2}T \rangle =(\Delta T)^{2}\). Such a relation between the SNR and the uncertainty in the estimation of a parameter at the QCRB has already been shown theoretically in [18, 58] and use in experiments to show the operation of an interferometer at the QCRB [59].

In our experimental procedure, we first calibrate the transmission modulation amplitude *δT* vs. the applied voltage to the EOM (see Appendix A.2). This then allows us to ramp down the calibrated transmission modulation amplitude *δT* to determine its value at a SNR = 1, marked with the circled “X” in Fig. 2. We finally take advantage of Eq. (22) to obtain the uncertainty in the estimation of transmission from the measurements for a given mean transmission *T* by taking the square of \(\delta T|_{\text{SNR}=1}\).

### 1.4 A.4 Photon counting

The dependence of the QCRB on the number of photons used to probe the systems under study makes it necessary to estimate such a quantity for the bTMSS, which is a continuous state. To go from a continuous photon flux to a discrete photon number requires bucketing the flux into discrete measurement time bins. The effective measurement time, *t*, for such a time bin is given by the response time of the measurement apparatus. In our experiments the measurement response is dominated by the spectrum analyzer, such that the effective measurement time is set by its RBW. To find the effective measurement time for a given RBW, we first find the relationship between the variance measured with the spectrum analyzer and the number variance of the detected optical state. We then take advantage of the known relationship between the number variance and mean photon number for the coherent state, that is \(\langle \Delta ^{2}\hat{n} \rangle = \langle \hat{n} \rangle \).

We consider the basic spectrum analyzer design shown in Fig. 6 to determine *t*. Since the output, *O*, of the spectrum analyzer is proportional to the variance of the input, *I*, we first determine the proportionality constant, *K*, between these two quantities. In the experiment, the value of this constant has no effect on the number of photons measured, as it is purely a scaling factor due to the electronics and therefore needs to be taken into account. To find the proportionality constant, we first consider a deterministic signal \(I_{\text{dtm}}=A\sin (2\pi ft+\phi )\) with variance \(\langle \Delta ^{2}I_{\text{dtm}} \rangle =A^{2}/2\). If we set the frequency of the electronic local oscillator (LO) used to demodulate the in phase (cos) and out of phase (sin) components of the input to the same frequency as our deterministic signal, we get – after splitting, LO mixing, resolution bandwidth filtering, squaring, and summing (see Fig. 6) – an output of the form

$$ \langle O_{\text{dtm}} \rangle =\frac{A^{2}}{8} \bigl\vert H(0) \bigr\vert ^{2}=K\frac{A^{2}}{2}, $$

(24)

where \(H(f)\) is the frequency response of the RBW filter. As a result,

$$ K=\frac{1}{4} \bigl\vert H(0) \bigr\vert ^{2}, $$

(25)

such that the ratio \(\langle O \rangle /K\) gives the actual variance of the input, \(\langle \Delta ^{2}I \rangle \). The number variance for an input state can then be related to the output of the spectrum analyzer through error propagation according to

$$ \bigl\langle \Delta ^{2}\hat{n} \bigr\rangle = \bigl\langle \Delta ^{2}I \bigr\rangle {\Big/} \biggl\vert \frac{\partial \langle I \rangle }{\partial \langle \hat{n} \rangle } \biggr\vert ^{2} =\frac{ \langle O \rangle }{K} {\Big/} \biggl\vert \frac{\partial \langle I \rangle }{\partial \langle \hat{n} \rangle } \biggr\vert ^{2}. $$

(26)

Next, we consider an input state given by a coherent state to take advantage of the relation between the number variance and the mean number of photons to calculate the effective integration time *t*. To do so we need to calculate the different terms on the right-hand-side of Eq. (26). The mean value for an input coherent state is given by

$$ \langle I_{\text{coh}} \rangle =C_{p\rightarrow i} \vert \alpha \vert ^{2}=C_{p \rightarrow i}\frac{ \langle \hat{n} \rangle }{t}, $$

(27)

where \(C_{p\rightarrow i}\) is the gain of the photodetector, \(|\alpha |^{2}\) is the mean photon flux of the coherent state, and \(\langle \hat{n} \rangle \) is the average number of photons detected over measurement time *t*. The mean value of the output of the spectrum analyzer in this case corresponds to the noise power of the coherent state. The expectation value of the output can be shown to be given by

$$ \langle O_{\text{coh}} \rangle = \frac{ \vert \alpha \vert ^{2}C_{p\rightarrow i}^{2}}{2} \int _{-\infty}^{\infty } \bigl\vert H(f) \bigr\vert ^{2}\,df. $$

(28)

As a result, the variance of the input takes the form

$$ \bigl\langle \Delta ^{2}I_{\text{coh}} \bigr\rangle = \frac{ \langle O_{\text{coh}} \rangle }{K}=2 \vert \alpha \vert ^{2}C_{p \rightarrow i}^{2} \frac{\int _{-\infty}^{\infty } \vert H(f) \vert ^{2}\,df}{ \vert H(0) \vert ^{2}}. $$

(29)

We can see that Eq. (28) is proportional to \(\int _{-\infty}^{\infty } \vert H(f) \vert ^{2}\,df\), while the output from a deterministic modulation input, Eq. (24), is proportional to \(\vert H(0) \vert ^{2}\). This difference comes from the deterministic signal being a single frequency peak, such that the contribution to the power spectrum (output of spectrum analyzer) is dominated by the power in the deterministic signal. On the other hand, the intensity noise of the coherent state is broadband, such that the power is distributed over all frequency components of the system response.

To find the measurement time, *t*, we can use Eqs. (27) and (29) in Eq. (26) to find the number variance for a coherent state

$$\begin{aligned} \bigl\langle \Delta ^{2}\hat{n} \bigr\rangle _{\text{coh}} =&2 \langle \hat{n} \rangle t \frac{\int _{-\infty}^{\infty } \vert H(f) \vert ^{2}\,df}{ \vert H(0) \vert ^{2}} \end{aligned}$$

(30)

$$\begin{aligned} =& \langle \hat{n} \rangle , \end{aligned}$$

(31)

where the last line is a property of the coherent state. We can now set the right-hand-side of Eq. (30) equal to Eq. (31) to show that the effective measurement time is given by

$$ t= \frac{ \vert H(0) \vert ^{2}}{2\int _{-\infty}^{\infty } \vert H(f) \vert ^{2}\,df }. $$

(32)

For an ideal spectrum analyzer, the RBW filter has a Gaussian profile with the full-width at half-maximum given by the value of the RBW, such that

$$ \frac{ \vert H_{\text{gaus}}(f) \vert ^{2}}{ \vert H_{\text{gaus}}(0) \vert ^{2}}=\exp\left(- \frac{4\ln (2)\:f^{2}}{\text{RBW}^{2}}\right). $$

(33)

Therefore, the time for our measurements would ideally be

$$ t_{\text{gaus}}=\sqrt{\frac{\ln (2)}{\pi}}\frac{1}{\text{RBW}}\approx \frac{0.47}{\text{RBW}}. $$

(34)

However, real spectrum analyzers are only able to approximate a Gaussian filter. For our spectrum analyzer (Agilent model E4445A) the filter is a 4-pole synchronously tuned filter with a correction factor of ≈ 0.94 [60]. Thus, the actual effective measurement time for our system is given by \(t\approx 0.44/\text{RBW}\).

Finally, to obtain the number of photons used in the experiment, we need to calibrate the photon flux, \(\Phi = \langle \hat{n} \rangle /t\), for the probe power used in the experiment. The DC voltage output, \(V_{\text{dc}}\), of the photodetectors used to detect the probe and conjugate modes are linearly dependent on the optical power, *P*, detected, such that \(V_{\text{dc}}=mP\) with *m* giving the proportionality constant. We first calibrate *m* by flipped a power meter in and out of the beam path in front of the photodetector and measuring the optical power *P* and output voltage \(V_{\text{dc}}\) for incident beams with different powers. This allows us to perform a linear fit to get an accurate measure of *m*. Thus, we can find the photon flux according to \(P=\Phi hc/\lambda \), where *λ* is the wavelength of the probe mode (795 nm for our experiment), *h* is Planck’s constant, and *c* is the speed of light. The photon number is then given by

$$ \langle \hat{n} \rangle =\Phi t=\frac{\lambda}{hc} \frac{t}{m}V_{\text{dc}}, $$

(35)

where \(V_{\text{dc}}\) is recorded for each transmission for each data set measured to determine the mean transmission *T* independently for each data point taken. The number of probing photons is calculated by bypassing the system under study and is also measured for each data point taken.

### 1.5 A.5 Inferring the squeezing parameters

The effective squeezing parameter, *s*, and the transmission of the probe through the Rb cell, \(T_{a}\), must both be estimated to find the state generated. The probe transmission measured without the pump field is not an accurate estimation of \(T_{a}\) as the strong pump field leads to optical pumping, which modifies the transmission of the probe. To estimate *s* and \(T_{a}\) we measure the noise properties of the generated bTMSS and compare the measured values with the theoretically calculated noise properties that take into account the distributed losses in the atomic medium.

For the generated bTMSS, we measure the balanced intensity-difference noise and the individual intensity noises of the probe and conjugate modes at 1.5 MHz, the operating frequency of the experiment, and subtract the electronic noise. These noises are then normalized by the corresponding shot noise and backtracked to obtain the normalized noises generated by the FWM process. We backtrack the noises by removing the effects of the loss from the Rb cell output window, optical path to the detectors, and the quantum efficiency of the detectors. This is done by using the relation

$$ \text{N}_{0}=\frac{\text{N}_{\text{m}}-(1-\eta )}{\eta}, $$

(36)

where N_{0} is the normalized noise directly generated by the source, N\(_{\text{m}}\) is the measured normalized noise, and *η* is the total transmission, which is given by \(T_{\text{p}}\eta _{\text{p}}\) for the probe beam and \(\eta _{\text{c}}\) for the conjugate beam. As described above, both transmission are the same so we can take \(\eta =\eta _{\text{c}}=T_{\text{p}}\eta _{\text{p}}\) to backtrack the intensity-difference noise.

To calculate the theoretically expected noise properties that take into account distributed losses in the source, we model the FWM in the atomic system as an infinite series of infinitesimal layers of two-mode squeezers and beam splitters (to take into account distributed losses) [41, 53]. Given that the frequency of the conjugate beam is far away from any atomic resonance, we assume that it does not experience any losses. Thus, \(T_{a}\) represents the loss due to atomic absorption for the probe beam. The sum of all the infinitesimal squeezing parameters gives the effective value of *s* and the product of all the infinitesimal transmissions of the beam splitters gives \(T_{a}\). For the theoretical normalized noises, we use the results given in [53] and verify them through the numerical approach given in [41]. The analytical solutions for the theoretical model for the normalized noises are given by

$$\begin{aligned}& \frac{ \langle \Delta ^{2} (\hat{n}_{p} - \hat{n}_{c} ) \rangle }{ \langle \hat{n}_{p} \rangle + \langle \hat{n}_{c} \rangle } = 1- \frac{2s\sinh ^{2} \left(\frac{\xi}{4} \right)}{\xi \cosh \left(\frac{\xi}{2}+\zeta \right)} -\sqrt{T_{a}} \frac{s\ln ^{2}(T_{a})\sinh ^{4} \left(\frac{\xi}{4} \right)}{2\xi ^{3}\cosh \left(\frac{\xi}{2}+\zeta \right)} , \end{aligned}$$

(37)

$$\begin{aligned}& \frac{ \langle \Delta ^{2}\hat{n}_{p} \rangle }{ \langle \hat{n}_{p} \rangle }= \frac{16s^{2} \left\{1-\sqrt{T_{a}} \left[1-\cos \left(\frac{\xi}{2} \right) \right] \right\}+\ln ^{2}(T_{a})}{\xi ^{2}} , \end{aligned}$$

(38)

$$\begin{aligned}& \frac{ \langle \Delta ^{2}\hat{n}_{c} \rangle }{ \langle \hat{n}_{c} \rangle } = \frac{16s^{2}\sqrt{T_{a}}}{\xi ^{2}}-1 - \frac{2\sqrt{T_{a}} \left[ (8s^{2}-\xi ^{2} )\cosh \left(\frac{\xi}{2} \right)+\xi \ln (T_{a})\sinh \left(\frac{\xi}{2} \right) \right]}{\xi ^{2}}, \end{aligned}$$

(39)

where \(\xi =\sqrt{16s^{2}+\ln ^{2}(T_{a})}\) and \(\tanh (\zeta )=\ln (T_{a})/\xi \).

We then determine *s* and \(T_{a}\) by finding the values of these parameters that provided the best match between the theoretical model and measurements, in log scale, through the goodness-of-fit parameter \(\chi ^{2}\) [61],

$$ \chi ^{2}=\sum_{i} \frac{ [\text{Measurement}_{i}-\text{Theory}_{i}(s,T_{a}) ] ^{2}}{\text{Variance of Measurement}_{i}}, $$

(40)

where the sum is over the three normalized noises. We minimize \(\chi ^{2}\) using a differential evolution optimization algorithm [62, 63] to find \(s=2.04\) and \(T_{a}=71\%\) with a \(\chi ^{2}=0.4563\). The uncertainty in the fitted parameters is found by varying their values until \(\chi ^{2}\) increases by the reduced \(\chi ^{2}\). The reduced \(\chi ^{2}\) is \(\chi ^{2}/\text{dof}\), where dof represents the degrees of freedom given by the number of measurements minus the number of parameters to fit. For our case the dof is 1, so \(\chi ^{2}/\text{dof}=\chi ^{2}\). Thus, we determine the values of the parameters for which \(\chi ^{2}\) is doubled to find the uncertainties for the parameters. Altogether, we find \(s=2.04\pm 0.02\) and \(T_{a}=71\%\pm 2\%\).

The optimization algorithm we used performs a differential evolution, which is a type of genetic algorithm. As such, it starts with a random set of possible solutions, tests how well each solution works using some goodness-of-fit parameter, and then mixes the solutions in an attempt to increase the goodness-of-fit. Here, minimizing \(\chi ^{2}\) serves as our goodness-of-fit. To initialize the algorithm, we randomly created a population with 5000 points of *s* and \(T_{a}\) values limited to \(0\le s\le 3\) and \(0.5\le T_{a}\le 1\). We then record the \(\chi ^{2}\) value for each of these points.

After initializing the population, we mix the different elements in a way that optimizes towards the lowest \(\chi ^{2}\) value. To do this we first find the optimal point, \(P_{o}\), that has values of *s* and \(T_{a}\) that give the lowest \(\chi ^{2}\) of all the points in the population. We take this point to make new points for the next generation of population. We select one of the remaining 4999 points to possibly replace, \(P_{i}\). To create the possible replacement point, \(P_{r}\), we randomly select two more points from the current population, \(P_{j}\) and \(P_{k}\), such that \(P_{o}\ne P_{i}\ne P_{j}\ne P_{k}\). We then create a vector pointing from \(P_{j}\) to \(P_{k}\) normalized by the limits placed on the parameters, that is, normalized by \(\sqrt{(0-3)^{2}+(0.5-1)^{2}}\) for our case. We create the replacement point \(P_{r}\) by adding this vector to \(P_{o}\). If the new point is outside of the limits set for *s* and \(T_{a}\) for one or both of the parameter values, the point is set to the closest limit. We then find \(\chi ^{2}\) for \(P_{r}\) and, if it is lower than the \(\chi ^{2}\) of point \(P_{i}\), we replace \(P_{i}\) with \(P_{r}\). Finally, we repeat this for all points that are not \(P_{o}\), randomly picking new \(P_{j}\) and \(P_{k}\) for each one. To increase the algorithm’s ability to find global minimums in the presence of local minimums, we perform the replacement for only 70% of the time when \(P_{r}\) is better than \(P_{i}\).

There is no set limit to the number of iterations needed to find the minimum \(\chi ^{2}\), so we iterate the algorithm until the spread of the population is orders of magnitude less than the uncertainty in the parameter values. Saving the \(\chi ^{2}\) value for each point during the differential evolution gives a look at the \(\chi ^{2}\) dependence on *s* and \(T_{a}\), especially around the global minimum. Thus, we can find a circular region around the minimum \(\chi ^{2}\) where its value is doubled to find the uncertainty in the transmission and squeezing parameter.