Engineering of two-photon superposition states

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Homodyne detector

Homodyne detector is devised to provide the measurement of a single-mode quadrature xθ for characterization of an optical quantum state or conditional preparation of quantum states. We will see that the Wigner function of its POVM element is Gaussian, thus the homodyne detector can be categorized as a Gaussian detector [22].

Basic properties

The scheme is illustrated in figure 3.2. A signal under investigation ˆρs is mixed with a classical field |αi (local oscillator ) by a balanced (50/50) beam-splitter 6. The phase of the local oscillator is adjusted by a PZT for accessing quadrature values with different phases θ. The two interfering modes after the beam splitter are then detected by a pair of identical photodetectors (typically linear photodiodes). Finally the difference of the generated photocurrents is obtained by an electronic subtraction. This difference is directly linked to the field quadrature with a scaling factor.

Conditional state preparation

A unique property of quantum measurement is that it allows to modify states by appropriately choosing a measurement strategy. This kind of method for state engineering is thus called conditional state preparation for highlighting the probabilistic but heralded character of this technique.

Composite system

Conditional preparation of a quantum state always involves at least two-mode composite system, the conditional mode and the signal mode. In particular, measurement is made on one part of a bipartite correlated system; the action of this measurement is to project the other part to a target state as shown in figure 3.3.

Calculation based on density matrix

Generally, it is not easy to analytically calculate the non-unitary beam-splitter operator, especially in the case of mixed states. An alternative way for obtaining the conditional state is to truncate the involved states into a finite-dimension Hilbert space, and then to use matrix manipulation and simple linear algebra.
Therefore, the first step is to represent all the involved quantum states, quantum channels and quantum measurements by density matrices with truncated Fock state dimensions. As shown in equation (1.46), the beam-splitter operator on the two-mode Fock state results in ˆB |n1, n2i = X N1,N2 Bn1,n2 N1,N2 |N1,N2i .

MaxLik for quantum state tomography

Quantum state tomography is a technique for extracting the full information of a quantum state by subjecting it to an ensemble of quantum measurements. As we know, a quantum state can be completely described by its density matrix (or equivalently by its Wigner function). For quantum optics in free space, the reconstruction of quantum sates is usually implemented by a set of quadrature values from homodyne measurements. In this case, the measurements of the quadrature ˆxθ give the marginal distribution P(xθ) = ∫ dyθW(xθ, yθ).
The next step is to reconstruct the Wigner function W. There exit several algorithms for quantum tomography [59], such as inverse Radon transformation, pattern functions, and Maximum-likelihood algorithm (MaxLik algorithm) . The most commonly used technique henceforth is the MaxLik algorithm due to three main advantages [62]. First, MaxLik algorithm enables us to compensate the optical losses; second, it allows us to incorporate the positivity and unity-trace constraints into the reconstruction procedure, thus always leading to a physically state; third, it gives the highest accuracy (with an intrinsic numerical noise defined by the Cramer-Rao bound). Nevertheless, Max- Lik algorithm also has some limitations, such as the truncation of the Hilbert space and the requirement of more computational resources. Here we will give a brief introduction about this MaxLik algorithm, which has been extensively used during this PhD work. The measurement outcomes of the homodyne detection can be organized as {θj , xj , fj} where fj is the frequency of occurrence for each outcome {θj , xj}. The likelihood is then given by L(ˆρ) = Y j P(θj , xj)fj .

High-finesse cavity

Compared to a low-finesse cavity, locking a high-finesse cavity is more challenging due to the limited resolution of the ADC. For instance, for a finesse of 1000, the number of sampling points to cover the whole peak is around 4096/1000 ≃ 4 which is obviously insufficient to lock the cavity. To overcome this problem, one can combine two 12-bit ADCs in order to obtain a 24-bit ones as mentioned previously. Another way to address the problem is to use two different scanning modes, namely a long scan and a short scan, as explained now.
The flowchart of the program execution is given in figure 4.4. One output of the DAC (DAC1) is used for the long scan, which spans over more than a free spectral range in order to identify the rough peak position. Another output (DAC0) is used for the short scan around the peak position identified by the long scan mode and also used for the locking.
The two outputs are summed with different gains. Such uneven allocation of sampling over the free spectral range enables the full use of the DAC resolution. Since there are four DACs available in the microcontroller development board, this method can be easily implemented without the need of any additional electronic building.
Figure 4.5 gives an example of the process for locking a 0.5 mm-long cavity [38] with a finesse of 1000. Note that in order to make the re-lock process faster when the cavity is unlocked, the program goes firstly to the short-scan mode instead of directly using the long-scan one. If the short scan still cannot find the peak after 10 sweeps then the program jumps to the long-scan mode.

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Table of contents :

Abstract
Acknowledgment
Contents
Introduction
Publications and awards
I Theoretical and Experimental Tools 
1 Quantum Theory of Light 
1.1 Photons
1.2 Representations of quantum states
1.3 Gaussian manipulations
1.4 Gaussian states
1.5 Non-Gaussian states
1.6 Conclusion
2 Nonlinear Frequency Conversion 
2.1 Introduction
2.2 Spontaneous down-conversion
2.3 Coherent frequency up- and down-conversion
2.4 Conclusion
3 Quantum Theory of Measurements 
3.1 Positive Operator Valued Measures
3.2 Photon detectors
3.3 Homodyne detector
3.4 Conditional state preparation
3.5 MaxLik for quantum state tomography
3.6 Conclusion
4 Automatic Locking System 
4.1 Introduction
4.2 Algorithm and model
4.3 Cavity locking
4.4 Integration and remote monitor
4.5 Conclusion
II Hybrid Entanglement Generation 
5 Heralded Fock States 
5.1 Single-photon state generation
5.2 Two-photon state generation
5.3 Engineering of two-photon superposition states
5.4 Heralding photons with temporal separation
5.5 Conclusion
6 Schrödinger Cat States 
6.1 Generation of odd cat states
6.2 π-phase gate for generating even cat state
6.3 State engineering with time-separated conditioning
6.4 Conclusion
7 Hybrid entangled states 
7.1 Hybrid qubit entanglement
7.2 Hybrid qutrit entanglement
7.3 Additional subtraction for hybrid qubit entanglement
7.4 Squeezing-induced micro-macro states
7.5 Conclusion
III Frequency Up-conversion 
8 Coincident Frequency Up-conversion System 
8.1 Introduction
8.2 Synchronized fiber lasers
8.3 Up-conversion system
8.4 Results and discussion
8.5 Conclusion
9 Applications of Frequency Conversion System 
9.1 Infrared photon-number-resolving detection
9.2 Few-photon-level infrared imaging
9.3 Generation of mid-infrared light
9.4 Conclusion
Conclusion and outlook
Appendix
A Mathematical formula 
A.1 Gauss integrals
A.2 Laguerre polynomials
A.3 Hermite polynomials (physicists’ version)
B g(2)(0) invariance to loss 
B.1 Expectation value for symmetric ordered operator
B.2 Symmetric ordering of field operator
B.3 Single-mode Gaussian state
B.4 Loss on single-mode Gaussian state
B.5 g(2)(0) invariant to loss
C Homodyne data simulation 
C.1 Intuitive method
C.2 Homodyne signal simulation
C.3 Quantum state reconstruction
D Qmixer 
D.1 Fock states generation
D.2 Superposition of |0i and |2i
D.3 Photon-subtracted squeezed vacuum states
D.4 Squeezed cat states from |ni
D.5 Quantum-optical catalysis
D.6 Amplification of cat states
E MCU locking 
E.1 Pseudo code for multi-locking
E.2 Configuration of integration box
F SSPD 
F.1 Structure
F.2 Performance
F.3 Optimization
Bibliography

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