Homodyne detection as a projective measurement

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Single-mode OPO in a ring cavity

In order to predict the quantum state produced by a lossy below-threshold OPO, we use a simple model where a coherent state CW pump undergoes a degenerate parametric downconversion process inside an optical cavity at resonance with the signal eld. In the majority of the literature about OPOs and their below-threshold counterpart, the optical resonator is assumed to have low optical losses so that a rst order development of the round trip equation can be performed (see e.g [Morin 14a]). It is generally a good assumption for high nesse cavities. For optical cavities of lower nesse, this assumption no longer holds so we use the full cavity transfer function. We will compute the variance of the quadratures of the output state by using the input/output relations of the ring cavity presented on gure 2.2.

A technical description of our OPO

In this section, we recall and update the technical description of the optical cavity used for our multimode OPO. A complete description is given in [Medeiros de Araujo 12]. Minor changes have been made since mostly consisting in replacing some mirrors.
In a nutshell, the optical resonator is a ring cavity designed for S-polarized light. Its length is about 3:95m (L = c=fr ) so that every optical frequency of the frequency comb produced by our light source (see gure 2.1) is resonant. For stability reasons, the ring cavity is folded several times in a complex fashion. A picture of the cavity can be seen in appendix A.

Single-mode squeezing measurement

Despite the fact that an OPO pumped below threshold is inherently multimode, one may always try to measure the quantum state at the output by mixing it with a Local Oscillator (LO) in a homodyne detection scheme as presented in chapter 3. For this measurement, we use our laser source presented on gure 2.1 as a LO in a scheme shown on gure 2.7.
The noise power in the homodyne detection signal is detected at a single sideband frequency and acquired while the phase between the signal and the LO is swept. The measurement is shown on gure 2.7. It shows the signal alternating between below and above the shot noise power. The average squeezing measured is 􀀀2:78dB relatively to shot noise, not corrected from any losses. This result was obtained with diodes whose losses were at least 7% each due to poor quantum eciency. For information, the best squeezing obtained with the aforementioned diodes was 􀀀3:5dB. Some higher squeezing of 􀀀5:5dB has been measured with diodes of near unity quantum eciency (see [Cai 15]) in the same conguration.

Lossless multimode OPO

A multimode description of the OPO has been rst developed in [Jiang 12] and reformulated in [Medeiros de Araujo 12]. A description of the temporal properties of the output multimode quantum state can also be found in [Averchenko 11]. In order to provide a description accounting for the multimode aspect of an OPO below threshold, we assume that the optical resonator is lossless. Actually, it is possible to write the squeezing spectrum of a lossy cavity as long as losses are at over the optical bandwidth. The optical resonator presented on gure 2.2 is modied so that ro = 1 and the output eld becomes ˆ aL.

Degenerate parametric down-conversion in a cavity

The Synchronously Pumped Optical Parametric Oscillator (SPOPO) is an OPO pumped by a frequency comb. The pump comb is generated through the doubling of the laser source as explained in section 2.4.2. The original eld is a frequency comb of repetition rate !r and Carrier-Envelope-Oset (CEO) !ceo. The pump eld is thus a frequency comb of repetition rate !r and Carrier-Envelope-Oset (CEO) 2!ceo within the spectral envelope presented on gure 2.6. In our Type-I degenerate Parametric Down-Conversion (PDC), each individual frequency !p;i = i !r + 2!ceo of the pump comb undergoes parametric down-conversion and creates a pair of correlated photons at the optical frequencies of the signal allowed by the optical cavity as illustrated on gure 2.8. For a pump frequency !p;i , the possible signal frequencies are !s;n = n!r + !ceo and !s;m = m!r + !ceo withm + n = i so that the energy is conserved: !p;i = !s;n + !s;m.

Parametric down-conversion in BiBO

We use a 2mm thick BiBO crystal as a gain medium to achieve Type-I degenerate PDC in our OPO. The details of the crystal orientation and phase-matching are also presented in appendix B. The advantage of BiBO is that its high non-linear coecients allows to produce measurable signicant squeezing without the need of a high nesse cavity.
Assessing the phase-matching function PDC (!s ;!i ) gives some insight about the correlation (or anticorrelation) between the signal and idler photons, !s and!i being the frequencies of the signal and idler beam. Figure 2.9 shows the phase-matching function PDC (!s ;!i ) for such a crystal where the angle of BiBO has been set so that the phase-matching k is null for 0 = 795nm. Figure 2.9 reveals a strong anticorrelation between the signal and idler frequencies as expected from the conservation of energy. The actual range of possible phase-matching is much wider than the optical bandwidth depicted on gure 2.9 and is eventually limited by the decreasing value of the eective nonlinear coecient of BiBO [Ghotbi 04a]. We will assume a constant non-linear coecient (2) over our optical bandwidth.

Hamiltonian of the conversion

In order to write the Hamiltonian of the conversion process, we assume the pump beam to be a coherent state. The other states at play having no mean eld, we neglect the uctuations of the pump and keep only the mean eld of the pump. We therefore replace its frequency-dependent annihilation operator by its spectral prole u˜p in the expression oh the Hamiltonian. We also assume that the depletion of the pump is negligible which, in a weak conversion process, is a very good approximation. We denote ˆ ay n and ˆ ay m the creation operators at the signal frequencies !s;n and !s;m. The Hamiltonian of the PDC described above is then: ˆH PDC / i X n;m ˜p (!s;n + !s;m)PDC (!s;n;!s;m)aˆy n ˆ ay m + h.c.

Toward a full model of SPOPO

In [Jiang 12] and [Medeiros de Araujo 12], a gap is bridged between the description of the multimode parametric down-conversion presented above and the full interpretation of the SPOPO as a multimode OPO in the spectral domain. To reach that conclusion, it is necessary to assume that optical losses are frequency independent within the optical bandwidth of the phase-matching function PDC. This assumption leads to the spectral eigenmodes of the PDC to be a also eigenmodes of the propagation in the optical resonator. Those eigenmodes of the SPOPO as a whole are frequency combs possessing dierent spectral envelopes and we call them supermodes.
A symplectic description of the SPOPO shows that each of those supermodes interacts independently with the pump beam to create squeezed states. Similarly to an OPO, the squeezed states are hosted in the correlated sidebands of the comb teeth at frequencies inferior to the bandwidth of the optical resonator. The squeezing spectrum is computed in [Medeiros de Araujo 12]. Characterizing such a state is not straightforward are requires the ability to engineer the LO spectrum [Pinel 12] in order to reveal the full multimode nature of the SPOPO output as we will see in the next chapter.
Finally, the theory developed previously fails to account for frequency-dependent losses in the resonator due to limited optical bandwidth and dispersion. We remain condent that this enrichment of the theoretical description may be within reach by adding frequency dependent losses and phase in equation (2.19).

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Scattering by the mask

Even if the phase mask composed of liquid crystals is fairly transparent, the presence of pixels along with the edges of the mask constitute a source of scattering. If the input light transverse prole is a well collimated Gaussian, the output prole may contain light in some Fourier components that were previously empty. When the pulse-shaper is paired with a spatial lter to achieve amplitude shaping, the lter lets only the 1 order of diraction pass. Some scattered light may still pass through the lter even for a grating amplitude A(!) = 0. The technique is eventually not exactly zero-background and the weak scattered light passing trough the lter may become comparable to the signal for low values of A(!).

Output beam geometry

The folded optical compressor composing the pulse-shaper separates a collimated beam composed of many wave vectors k(!) traveling in the same direction, rst angularly, then spatially. Those dierent optical frequencies must be recombined in a collimated beam at the output. If the distance between the cylindrical mirror and the SLM is not perfectly f , some geometric problems may appear in the output beam. The rst problem is that the output beam will not be collimated. This distance can be roughly adjusted with the laser source operating CW until the output beam is suciently collimated.
Then, one must check that the dierent wave vectors k(!) are recombined in the same direction. If not, the dierent wave vectors associated to dierent optical frequencies propagate in dierent direction. This phenomenon is known in ultrafast optics as angular chirp and is commonly created by dispersive optical elements. It can be measured with a spectrometer placed after an optical telescope by moving a pinhole located near the Fourier plane and monitoring any change of spectrum. Some angular chirp in the horizontal direction generally comes from a slight mismatch of the distance between the cylindrical mirror and the SLM. For angular chirp in the vertical direction, the grating should probably be adjusted to make sure its lines are perfectly perpendicular to the plane of the input beam.

Table of contents :

I Context and Tools 
1 Not an Introduction to Multimode Quantum Optics 
1.1 Elements of quantum optics
1.1.1 The quantized electric eld
1.1.2 Quantum states of interests
1.1.3 The density matrix
1.1.4 The Wigner function
1.2 Multimode generalization
1.2.1 Optical modes
1.2.2 Multimode electric eld
1.2.3 Multimode quantum states
1.2.4 Multimode density matrix
1.2.5 Multimode Wigner function
1.3 Multimode Gaussian states
1.3.1 Single-mode Gaussian state
1.3.2 The covariance matrix
1.3.3 Multimode squeezed vacuum
2 A Source of Spectrally Multimode Quantum States 
2.1 A crash course of ultrafast optics
2.1.1 Generation of ultrafast pulses
2.1.2 Pulse modeling
2.2 Our light source
2.3 Single-mode OPO in a ring cavity
2.3.1 Input/output relations
2.3.2 OPO threshold
2.3.3 Predicting squeezing
2.4 Measuring single-mode squeezing
2.4.1 A technical description of our OPO
2.4.2 Pump spectrum
2.4.3 Single-mode squeezing measurement
2.5 Lossless multimode OPO
2.5.1 Degenerate parametric down-conversion in a cavity
2.5.2 Parametric down-conversion in BiBO
2.5.3 Hamiltonian of the conversion
2.5.4 Eigenmodes of the conversion
2.5.5 Toward a full model of SPOPO
3 Tunable Projective Measurements 
3.1 Homodyne Detection
3.1.1 Single-mode homodyne detection
3.1.2 Homodyne detection as a projective measurement
3.2 Ultrafast Pulse Shaping
3.2.1 Diraction-based amplitude shaping
3.2.2 Pulse-shaper design
3.2.3 Potential problems
3.2.4 Dual beam pulse-shaping
3.3 Measuring multimode squeezed vacuum
3.3.1 Principle
3.3.2 Measurement
3.3.3 The covariance matrices
3.3.4 Eigenvalues and eigenmodes
3.3.5 Going further in multipartite entanglement
II Single-Photon Subtraction 
4 A Theoretical Framework for Multimode Single-Photon Subtraction 
4.1 Modeling single-photon subtraction
4.1.1 Detecting a single photon
4.1.2 Single-mode case
4.1.3 General multimode case
4.1.4 Application to multimode squeezed vacuum
4.2 Photon subtraction from spectrally/temporally multimode light
4.2.1 Linear and non-linear photon subtraction
4.2.2 Time-resolved detection of a photon
4.2.3 Single-photon subtraction kernel
5 Single-Photon Subtraction via Parametric Up-Conversion 
5.1 Theory of sum-frequency
5.1.1 Modes of the process
5.2 Collinear SFG in BiBO
5.2.1 Phase-matching
5.2.2 Choosing the crystal length
5.2.3 Playing with the gate
5.3 Non-collinear SFG in BiBO
5.3.1 Eect of focusing
5.3.2 Eect of birefringence
5.3.3 The problem of gate SHG
5.3.4 Eigenmodes of non-collinear subtraction
III Process Tomography 
6 Tomography of the single-photon subtractor 
6.1 Quantum process tomography
6.1.1 Tomography of a quantum black box
6.1.2 Checking the quantum process
6.2 Probing the subtraction matrix
6.2.1 The probing basis
6.2.2 Accessing the elements of the subtraction matrix
6.2.3 Experimental subtraction matrices
6.3 Maximum-Likelihood reconstruction
6.3.1 Principle of Maximum-Likelihood reconstruction
6.3.2 Optimizing with evolutionary strategies
6.3.3 Reconstruction in the pixel basis
6.4 Tomography in a natural basis
6.4.1 The measurement basis
6.4.2 Raw measurement data
6.4.3 Maximum-Likelihood reconstruction
Conclusion & outlooks
A The SPOPO cavity 
B Non-linear optics in BiBO 
C The fantasy of type II parametric interaction for QPG 


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