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Continuously tunable modulation scheme for homodyne detection and state tomography

Benjamin Bret Lane, Junxin Chen, Danial Shadmany, Robert Lanza, and Nergis Mavalvala

Open Access Open Access

Abstract

Homodyne detection is a common self-referenced technique to extract optical quadratures. Due to ubiquitous fluctuations, experiments measuring optical quadratures require homodyne angle control. Current homodyne angle locking techniques only provide high quality error signals in a span significantly smaller than π radians, the span required for full state tomography, leading to inevitable discontinuities during full tomography. Here, we present and demonstrate a locking technique using a universally tunable modulator which produces high quality error signals at an arbitrary homodyne angle. Our work enables continuous full-state tomography and paves the way to backaction evasion protocols based on a time-varying homodyne angle.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Homodyne detection, among the most widely-used techniques in optics, allows one to measure a chosen quadrature in optical phase space. It is deployed in quantum sensing applications of gravitational wave measurement [1], quantum tomography [2], entanglement-based quantum key distribution systems [3], and other quantum information tasks to characterize the sideband fluctuations as a function of quadrature. Homodyne detection involves combining a signal with a phase-coherent reference at the same frequency – the local oscillator (LO) – at a beamsplitter and measuring the photocurrent generated by the light at the output ports. Balanced homodyne detection (BHD) is a homodyne technique based on measuring the difference of the photocurrents from both output ports, which suppresses classical noises that are common in the signal and the LO paths [4,5,6].

The measured quadrature of the homodyne detector depends upon precise control of the relative phase between the signal and the LO electric fields – i.e. the homodyne angle $\theta$. Varying the homodyne angle across a range of $\pi$ radians is needed to measure arbitrary quadratures and to fully tomographically reconstruct the state of the light. Fluctuating signal and LO path lengths lead to homodyne angle uncertainties, requiring control for a fixed homodyne angle. A traditional method for controlling the homodyne angle involves using the linear portion of the sinusoidal homodyne fringe as an error signal [7]. Due to the sinusoidal nature of these error signals, they are only approximately linear within the interval $\pm \pi /4$, outside which the quality of the error signal drops significantly. To get a high quality error signal close to the top or bottom of the fringe, one typically modulates either the signal or the LO electric field in purely amplitude or phase to get an error signal proportional to the derivative of the fringe of the homodyne angle, however the quality of this error signal is then poor in the linear part of the fringe. Thus to perform full state tomography, one needs to change the locking protocol, often breaking the lock.

In this paper, we propose and demonstrate a free-space scheme to produce a high quality error signal to control the homodyne angle, which works at arbitrary homodyne angles using a single universally tunable modulator (UTM) [8]. This control technique utilises the fact that the UTM can produce modulation of light at arbitrary angles in the amplitude-phase plane. Similar continuous quadrature modulation techniques have been successfully applied to classical optical communication protocols to perform optical carrier recovery and to reduce phase noise in the LO [9,10]. Using a UTM, our scheme enables continuously controlling the homodyne angle between arbitrary start and end points without the loss of lock, extending the toolbox of quantum tomography and paving the way for full state tomography of short lived quantum states in one shot. One potential application for this novel, continuous locking scheme is in realizing the backaction evasion state estimation scheme proposed by Miao et. al. in [11].

2. Theory

In this section, we analyze the error signal for balanced homodyne detection provided by our scheme using a formalism based on [8]. For general homodyne detection, the error signal can be derived similarly.

Figure 1(a) shows a schematic of the universally tunable modulator. The amplitude and phase modulation quadratures $\tilde {A}$ and $\tilde {P}$ generated by the UTM after the polarizer are

$$\begin{aligned} \tilde{A} = \frac{\delta_1}{2}\cos(\frac{\sigma}{2})(1+\delta_re^{i\varphi}) \\ \tilde{P} = \frac{\delta_1}{2}\sin(\frac{\sigma}{2})(1-\delta_re^{i\varphi}) \end{aligned}$$
where $\delta _r=\delta _2/\delta _1$ is the ratio of the amplitudes of the two drives at the same frequency $\omega /2\pi$, and $\varphi =\varphi _1-\varphi _2$ is the relative phase of the drives ($\delta _j$ and $\varphi _j$ with $j=1,2$ are defined in the caption of Fig. 1). $\sigma$ is an angle related to input polarization, defined by $\mathrm {tan}^2(\sigma /2)=P_\mathrm {h}/P_\mathrm {v}$, where $P_\mathrm {h}$ and $P_\mathrm {v}$ are powers in horizontal and vertical polarization components of the input beam respectively. Sending circularly polarized input light ($\sigma =\pi /2$) is a typical way of operating the UTM. The modulation induces positive and negative sidebands $E_+$ and $E_-$ onto the LO light, with a frequency separation $\omega /2\pi$ away from the carrier:
$$E_+{=} \frac{1}{2}(\tilde{A}+i\tilde{P})$$
$$E_-{=} \frac{1}{2}(\tilde{A}^*+i\tilde{P}^*)$$

 figure: Fig. 1.

Fig. 1. Experimental setup. (a) UTM schematic for arbitrary quadrature modulation [8]. The quarter wave plate is used to adjust the polarization of the input beam. The white arrows indicates the modulation axes of the electro-optical crystals, which are perpendicular to each other, and 45° to the vertical direction. The vertically polarized polarizer projects the modulations to the phase space of a single optical mode. $\tilde {\delta }_j=\delta _j e^{i\varphi _j}$ with $j=1,2$ and real $\delta _j$ and $\varphi _j$ are complex phasors of the drives to crystals. (b) Simplified layout of the experimental setup. SIG represents the light whose quadratures are to be measured. Arrows indicate signal flow.

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This modulation is encoded in the BHD photocurrent. The balanced homodyne error signal $E_\mathrm {err}^\mathrm {c}$ is then given by the cosine component of the demodulated photo current (derivation is available in the appendix):

$$ E_\mathrm{err}^\mathrm{c} = E_0(r^2-t^2)\textrm{Re}[\tilde{A}]-4rtE_0^2\beta\textrm{Re}[e^{i\theta}(E_+{+}E_-)]$$
$$\begin{aligned} =& E_0(r^2-t^2)\textrm{Re}[\tilde{A}]\\&-2rtE_0^2\beta\delta_1(\mathrm{cos}(\theta+\sigma/2)+\delta_\mathrm{r}\mathrm{cos}\varphi\mathrm{cos}(\theta-\sigma/2)) \end{aligned} $$
where $E_0$ is the amplitude of the LO electric field at the carrier frequency, $r$ and $t$ are the homodyne beamsplitter (HBS) reflectivity and tranmissivity, $\beta$ is the ratio of the amplitude of the signal electric field to the LO electric field, and $\theta$ is the homodyne angle. The first line of Eq. (3) is an offset independent of homodyne angle and vanishes for a 50:50 homodyne beam splitter. The second line provides an error signal sinusoidal in $\theta$. $\sigma$ introduces phase lead and delay, respectively, in the two terms in the parenthesis of the second line, while $\delta _\mathrm {r}$ and $\varphi$ controls the weight of the terms. To see more clearly the role of $\delta _\mathrm {r}$ and $\varphi$ in the error signal, we choose $r=t=1/\sqrt {2}$ and $\sigma =\pi /2$, then the error signal can be reduced to
$$ E_\mathrm{err}^\mathrm{c}={-}\frac{E_0^2\beta\delta_1}{2\sqrt{2}}\left((1+\delta_r\mathrm{cos}\varphi)\mathrm{cos}\theta+({-}1+\delta_\mathrm{r}\mathrm{cos}\varphi)\mathrm{sin}\theta\right). $$
When $\delta _\mathrm {r}\geq 1$ the error signal can be completely determined by $\mathrm {cos}\theta$ or $\mathrm {sin}\theta$ for $\delta _\mathrm {r}\mathrm {cos}\varphi =1$ or $-1$ respectively, resulting in a desired $\pi /2$ phase shift in the error signal between the two cases. For any $\delta _\mathrm {r}<1$, the tuning range is reduced. For any $\sigma \neq n\pi$ with $n\in \mathbb {Z}$, the effect of $\sigma$ is adding weighting factors to the $\mathrm {cos}\theta$ and $\mathrm {sin}\theta$ terms, leading to a varying error signal oscillating amplitude as $\varphi$ changes. Interestingly, $\sigma$ does not contribute to phase shifting of error signal. Therefore, the maximum error signal phase shift is $\pi /2$ for this scheme.

3. Experimental demonstration

The layout of the experiment is shown in Fig. 1(b). Light from a 1064 nm ND:YaG laser is incident on a beamsplitter (BS1), dividing into the LO and the signal paths. The light in the LO path passes through a quarter-wave plate (becoming nearly circularly polarized), through the UTM, then through a vertical polarizer before recombining with the signal path light at the homodyne beamsplitter. Both ports of the homodyne beamsplitter (HBS) are incident upon photodetectors and their photocurrents subtracted, mixed with the phase shifted UTM drive and low-passed to create the error signal. This error signal is then fed into a PID controller to actuate a piezoelectrically driven mirror in the LO path to control the relative path length.

Our UTM was constructed by modifying a New Focus 4140 electro-optic amplitude modulator to allow separate drives to the two crystals. The UTM inputs were driven with a 10 MHz sine wave generated by a function generator with an additional relative phase $\varphi$ added to input 2.

Figure 2 shows the modeled and measured error signals for a range of relative drive angles. For each $\varphi$, the homodyne angle is controlled across a range of lockpoints while keeping the other PID settings constant. The homodyne angle $\theta$ is determined from the DC output voltage $V_\mathrm {DC}$ of the locked balanced photodetector by $\mathrm {cos}\theta =V_\mathrm {DC}/V_\mathrm {a}$ with $V_\mathrm {a}$ the amplitude of the homodyne fringe. This is repeated across a subset of the continuous relative drive angle from $\varphi =0$, ideally pure amplitude modulation, to $\varphi =\pi$, ideally pure phase modulation. The zero voltage lockpoint spans a homodyne angle of $\pi /2$ and the lockable range spans a homodyne angle greater than $\pi$. The varying error signal amplitude is attributed to the elliptical effective polarization seen by the UTM crystals due to non-zero birefringence at DC. We account for this effect by letting $\sigma$ and $\delta _\mathrm {r}$ to be free parameters and determine them through fitting the blue data set. Other modelled curves are derived without free parameters.

 figure: Fig. 2.

Fig. 2. Error signals for a variety of UTM drive angles. Each data point is acquired by stably locking the homodyne and performing a statistical average on the photocurrents. Error bars are within the markers. The insert zooms in on one data point to show error bars.

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The modeled error signal shows excellent agreement with the experimental data across all UTM angles. The error bars in 2 are statistical, arising from the averaging of a time series of photocurrents. The error signal voltage is derived by demodulating the subtracted outputs of PD1 and PD2. The homodyne angle is derived from the subtracted outputs of PD1 and PD2 and is calibrated against a sweep of the full homodyne range. The uncertainty in homodyne angle ranges between 0.12 and 0.57 degrees, with a mean uncertainty of 0.26 degrees.

4. Conclusion

In conclusion, we have demonstrated the use of a single universally tunable modulator to produce a high quality error signal at arbitrary homodyne angles. This enables measurement of homodyne angles across a range greater than $\pi$ without breaking lock. With a typical homodyne angle uncertainty of approximately 0.26 degrees, the UTM can in principle enable the straightforward high-resolution full tomography of short-lived quantum states and quantum states with angular features much smaller than those typically produced experimentally today.

Funding

National Science Foundation (PHY-2012088).

Acknowledgments

The authors thank MathWorks, Inc. for their computing support.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.

Supplemental document

See Supplement 1 for supporting content.

References

1. M. Tse, H. Yu, N. Kijbunchoo, et al., “Quantum-enhanced advanced ligo detectors in the era of gravitational-wave astronomy,” Phys. Rev. Lett. 123(23), 231107 (2019). [CrossRef]  

2. M. G. Raymer and M. Beck, 7 Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling (Springer Berlin Heidelberg, 2004), pp. 235–295.

3. W.-B. Liu, C.-L. Li, Y.-M. Xie, C.-X. Weng, J. Gu, X.-Y. Cao, Y.-S. Lu, B.-H. Li, H.-L. Yin, and Z.-B. Chen, “Homodyne detection quadrature phase shift keying continuous-variable quantum key distribution with high excess noise tolerance,” PRX Quantum 2(4), 040334 (2021). [CrossRef]  

4. M. Collett, R. Loudon, and C. Gardiner, “Quantum theory of optical homodyne and heterodyne detection,” J. Mod. Opt. 34(6-7), 881–902 (1987). [CrossRef]  

5. U. Leonhardt and H. Paul, “Measuring the Quantum State of Light,” Prog. Quantum Electron. 19(2), 89–130 (1995). [CrossRef]  

6. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009). [CrossRef]  

7. L. Kazovsky, “Balanced phase-locked loops for optical homodyne receivers: Performance analysis, design considerations, and laser linewidth requirements,” J. Lightwave Technol. 4(2), 182–195 (1986). [CrossRef]  

8. B. J. Cusack, B. S. Sheard, D. A. Shaddock, M. B. Gray, P. K. Lam, and S. E. Whitcomb, “Electro-optic modulator capable of generating simultaneous amplitude and phase modulations,” Appl. Opt. 43(26), 5079–5091 (2004). [CrossRef]  

9. Z. Liu, J.-Y. Kim, D. S. Wu, D. J. Richardson, and R. Slavík, “Homodyne ofdm with optical injection locking for carrier recovery,” J. Lightwave Technol. 33(1), 34–41 (2015). [CrossRef]  

10. T. Kan, K. Kasai, M. Yoshida, and M. Nakazawa, “42.3 tbit/s, 18 gbaud 64 qam wdm coherent transmission over 160 km in the c-band using an injection-locked homodyne receiver with a spectral efficiency of 9 bit/s/hz,” Opt. Express 25(19), 22726–22737 (2017). [CrossRef]  

11. H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, and Y. Chen, “Probing macroscopic quantum states with a sub-heisenberg accuracy,” Phys. Rev. A 81(1), 012114 (2010). [CrossRef]  

Supplementary Material (1)

NameDescription
Supplement 1       Derivation of error signal for UTM lock.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.

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Figures (2)
Fig. 1.
Fig. 1. Experimental setup. (a) UTM schematic for arbitrary quadrature modulation [8]. The quarter wave plate is used to adjust the polarization of the input beam. The white arrows indicates the modulation axes of the electro-optical crystals, which are perpendicular to each other, and 45° to the vertical direction. The vertically polarized polarizer projects the modulations to the phase space of a single optical mode. $\tilde {\delta }_j=\delta _j e^{i\varphi _j}$ with $j=1,2$ and real $\delta _j$ and $\varphi _j$ are complex phasors of the drives to crystals. (b) Simplified layout of the experimental setup. SIG represents the light whose quadratures are to be measured. Arrows indicate signal flow.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Error signals for a variety of UTM drive angles. Each data point is acquired by stably locking the homodyne and performing a statistical average on the photocurrents. Error bars are within the markers. The insert zooms in on one data point to show error bars.

View in Article | Download Full Size | PDF

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

A ~ = δ 1 2 cos ⁡ ( σ 2 ) ( 1 + δ r e i φ ) P ~ = δ 1 2 sin ⁡ ( σ 2 ) ( 1 − δ r e i φ )
E + = 1 2 ( A ~ + i P ~ )
E − = 1 2 ( A ~ ∗ + i P ~ ∗ )
E e r r c = E 0 ( r 2 − t 2 ) Re [ A ~ ] − 4 r t E 0 2 β Re [ e i θ ( E + + E − ) ]
= E 0 ( r 2 − t 2 ) Re [ A ~ ] − 2 r t E 0 2 β δ 1 ( c o s ( θ + σ / 2 ) + δ r c o s φ c o s ( θ − σ / 2 ) )
E e r r c = − E 0 2 β δ 1 2 2 ( ( 1 + δ r c o s φ ) c o s θ + ( − 1 + δ r c o s φ ) s i n θ ) .
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