1. Introduction

Engineering of arbitrary mesoscopic quantum states of light is a challenging task. Impressive results were already obtained using giant enhancement in superconducting cavities [1, 2], and protocols were proposed to generate arbitrary states with such systems [3], but the trapped state cannot be used for quantum communication protocols. In the case of free space propagating quantum states of light, the most common method for optical state engineering is to generate the state directly, by using two entangled beams and by performing a measurement on one of these, either by click counting [4, 5] or by homodyning [6–8]. Schrödinger cat states of light for instance, consisting in a coherent superposition of two coherent sates and composing a basic resource for quantum information processing, have been produced using these techniques [9, 10].

Building a state step by step is however necessary in order to grow its size, as the abovementioned methods are highly inefficient for producing large output states. Some protocols based on photon addition [11] or subtraction [12] propose this iteration of operations, but with the use of photon detection events which imply very low success probability. On the other hand, iterative generation based on homodyning has proven to be very efficient [13].

We propose here a generalization of the results presented in [13]: we present a setup for the generation and for the growth of arbitrary quantum states of light by the successive application of a simple protocol that will be described and explicitly calculated in a particular case in section 2, and whose performances will be discussed in section 3.

2. Protocol

The idea of the protocol is to build a superposition containing up to n + m photons by the “mixing” of two superpositions containing up to n and m photons. Let us first see the simple case where n = m = 1.

2.1. Simple case

The resource that we need to feed our protocol is the elementary superposition of vacuum and a single photon:

The principle is shown on Fig. 1: two of these resource states $|{\psi }_1〉=|{\psi }_1^{(1)}〉={\alpha }_1|0〉+{\beta }_1|1〉$ and $|{\psi }_2〉=|{\psi }_2^{(1)}〉={\alpha }_2|0〉+{\beta }_2|1〉$ are sent on a beamsplitter with transmission τ₁, and a homodyne detection is performed on one output arm. When the homodyne conditioning is successful (x′ = x′₁), the wavefunction of the other output arm state is projected on:

 

Fig. 1 Elementary protocol for generation and growth of arbitrary states: two states $|{\psi }_1^{(1)}〉$ and $|{\psi }_2^{(1)}〉$ of the form of Eq. (1) are mixed on a beamsplitter with transmission τ₁. A homodyne measurement is performed in one output arm, and the generation of a state |ψ_out〉 in the other arm is conditioned upon the homodyne event x′ = x′₁.

Download Full Size | PDF

Let us then remember the expression of the wavefunction of a Fock state $〈x|k〉=1/\left({\pi }^{1/4}\sqrt{2^{k}k!}\right)H_k(x)\mathit{exp}(-x^2/2)$, where H_k(x) is the k^th Hermite polynomial (of degree k): this form tells us that any superposition with up to n photons will have a wavefunction that will be written as a polynomial of degree up to n, times a gaussian of unit variance. This comes from the fact that the H_k polynomials are a basis of ℂ[X].

In the present case, Eq. (2) is the general writing of an arbitrary polynomial of degree up to two times a gaussian (all the polynomial can be split in ℂ[X]). According to the previous remark, this means that the corresponding state is an arbitrary superposition of up to two photons, whose parameters can be adjusted with α_i, β_i, τ₁ and x′₁.

2.2. General case

Let us generalize the idea of the previous paragraph by recurrence: let us suppose that we have been able to generate a superposition with up to n and m photons, and see how we can generate a superposition with n + m photons.

Mathematically speaking, it means that we assume to have created two states |ψ⁽ⁿ⁾〉 and |ψ^(m)〉 whose wavefunctions can be written as:

Let us mix these two states according to the same scheme of Fig. 1, by feeding the setup with |ψ⁽ⁿ⁾〉 and |ψ^(m)〉 instead of $|{\psi }_1^{(1)}〉$ and $|{\psi }_2^{(1)}〉$ (resp.). The state that we will thereby generate can be written as:

The protocol transformation is true for n = m = 1 and can be iterated for any n or m, which means that it is true for any n and m: we have proven that the use of the simple protocol of Fig. 1(a) iterated n times and fed by superposition of the form of Eq. (1) can generate arbitrary superpositions of up to n photons.

2.3. Practical calculation of the created state in a cascaded like configuration

Let us write the equations that have to be solved in order to find the proper parameters of the different beamsplitters and homodyne conditionings in a simple case. To simplify the calculations, let us consider a protocol which produces arbitrary superposition of n photons, by increasing the maximum number of photon one by one iteratively, according to the cascaded scheme shown on Fig. 2. We will suppose here that the weight of the component |n〉 is non zero (c_n ≠ 0) in the output state ${\sum }_{k=0}^n{c}_k|k〉$.

 

Fig. 2 Setup for the generation of arbitrary superposition of n photons, in a cascaded configuration.

Download Full Size | PDF

We suppose to have n input states of the form $|{\psi }_j^{(1)}〉={\alpha }_j|0〉+{\beta }_j|1〉$ at our disposal, and we mix them according to the scheme of Fig. 2. It is easy to check that the output state is of the form:

Equation (5) clearly shows that the output state is of the form of an arbitrary polynomial of degree n, times a gaussian of unit variance, which is the general writing of an arbitrary superposition of n photons: ${\sum }_{k=0}^n{c}_k|k〉$. The relation that fixes the conditions on the parameters α_i, β_i, τ_i and x′_i can finally be written as:

2.4. Structuration of the protocol

A great advantage of our protocol is that it allows for the use of quantum memories between each homodyne conditioning. These devices are currently developing very quickly [16], and they give a potential increase in the total success probability if the number of iterations increases. We have seen on Fig. 2 a possible configuration to generate an arbitrary superposition, but it is certainly not optimal, as we increase the degree of the polynomial by one at a time, and as all the input states are not used simultaneously. One possibility to optimize this is to perform a maximum of operations in parallel as possible, by “symmetrizing” the scheme. Figure 3(a) shows for instance one of these setups for n = 2^p. In this case, the degree of the polynomial is growing exponentially by advancing along the scheme. Quantum memories can then be naturally incorporated between successive stages: each time a homodyne conditioning succeeds, the created state is stored until the other required state is available.

 

Fig. 3 Advantage of the symmetrized configuration. (a) Setup of a symmetrized configuration, in the case where n = 2^p. The positions of quantum memories have been specified by the parenthesis symbols (). (b) Total success probability of a protocol involving eight input resource states given by Eq. (1) in a cascaded (solid blue line) and a symmetrical (dashed green line) configuration. The total success probability of a protocol not using any quantum memory is shown for comparison (dot-dashed red line).

Download Full Size | PDF

If one assumes, for instance, that all the homodyne conditionings have the same success probability P_mix, Fig. 3(b) shows the tremendous increase in the total success probability allowed by the use of quantum memories in a symmetrical protocol configuration, in the case where eight input resource states are used.

2.5. Example

Example 1: two photon superposition

Let us see with a concrete example how the protocol can be used to generate states of light, by studying how the protocol can output an arbitrary superposition of the form proposed in [12]:

First, given the expression of the target state, the weight of the two photon state is never 0, so we know that a₁a₂ ≠ 0. The two roots of the polynomial of the wavefunction of Eq. (2) are then b₁/a₁ and b₂/a₂. These should then be identified to the roots of the polynomial in the wavefunction of Eq. (9):

We can then first fix the transmission of the beamsplitter τ₁ and the homodyne conditioning x′₁ and find the proper parameters α_i and β_i of the input states to generate the desired output state.

Example 2: the three photon Fock state |3〉

As a second example, let us find a valid set of parameters for our setup to generate a three photon Fock state. As we have seen before we have more degrees of freedom than variables, so we can fix some of them. Let us then simplify the problem by replacing the input resource states of Eq. (1) by single photon Fock states (α₁ = α₂ = α₃=0), and by using a symmetrical beamsplitter for the first homodyne conditioning ( ${\tau }_1=1/\sqrt{2}$). Then, we have three variables (x′₁, x′₂ and τ₂) to find in our problem.

The new Eq. (8) to be solved can be written as:

This means that the mixing of two single photons on a symmetrical beamsplitter, with a conditioning on $x_1^{\prime }=\sqrt{3}/2$ produces a state which, mixed with another single photon on another symmetrical beamsplitter and a conditioning on x′₂ = 0, gives birth to a three photon Fock state.

Discussion

In these two examples, the user can adjust many parameters freely. For instance in the first example, the transmission τ₁ and the conditioning x′₁ were arbitrary, and in the second example the input states could be chosen to be single photon Fock states. This illustrates the remark formulated in the end of paragraph 2.3.

We will see in the next section how these could be explicitly chosen in order to maximize the success probability of the total operation in the case of a two photon Fock state superposition of Eq. (9), for which a graphical representation of the parameters is possible.

3. Performances

The performances of the protocol will be evaluated by calculating the fidelity of the state created by an imperfect setup with the ideal expected state. This mathematical tool, though it doesn’t guarantee the quality of the state (a high fidelity doesn’t necessarily mean a possibility to use the state in complex quantum protocols), yet gives a good idea of the robustness properties of the protocol.

3.1. Success probability

Obviously, heralding on events matching exactly the homodyne condition x′ = x′₁ will lead to a zero success probability, so one has to accept the events within a window x′ ∈ [x′₁ − Δx, x′₁ + Δx]. Increasing its width Δx will increase the success probability, but at the cost of a decrease in the quality of the state. In order to perform the study of the success probability of the protocol we propose to fix a target fidelity of the state we want to achieve, and to optimize the heralding width of the homodyne conditioning in order to maximize the success probability of the operation.

Let us first focus on the previous example of Eq. (9) in the case where c₀ = 0 and c₁ = 1: the state superposition 2^−1/2(|1〉 + |2〉). We have seen that the coefficients τ₁ and x′₁ could be freely chosen in order to generate it. By using Eq. (11), we can plot the success probability as a function of these two coefficients. Figure 4(a) shows this for a target fidelity of 90%, revealing that there is actually an optimal point for ( ${\tau }_1^2$, x′₁) around (0.32, 0.46) for the generation of the state, leading to almost 30% success probability of generation.

 

Fig. 4 Optimization of the success probability. (a) Success probability of the protocol for the generation of the state 2^−1/2(|1〉 + |2〉) as a function of the quadrature conditioning x′₁ = x′₀ and the energy transmission of the beamsplitter ${\tau }_1^2={\tau }^2$. The coefficients of the resource states are given by Eq. (11), and the target fidelity is 90%. (b) Optimized success probability for the states is shown as: Eq. (13) solid blue, Eq. (14) dashed red, Eq. (15) dot-dashed green and Eq. (16) solid thin black.

Download Full Size | PDF

This optimization can be performed on various states, showing some difference in the efficiency of production. For instance, Fig. 4(b) shows the optimized success probability as a function of the target fidelity for the four states of the form of Eq. (9):

We see that the success probability of our protocol is very high compared to other previously proposed setups. Indeed, the four states defined by Eqs. (13)–(16) were also studied in [12], and provided success probabilities of the order of 10⁻⁵ for target fidelities of 90%. In our case, these success probabilities are greater than 10% and reach almost 100% for the state of Eq. (16): this impressive behavior is simply explained by the fact that the unconditioned state (100% success probability by definition) has already 87% fidelity with the target state.

If we consider realistic success probability production of resource states of Eq. (1) of the order of 10⁻³ (success probability production of photon Fock state of the order of 1% [17] and conditioning rate of 10% [7]), the success probability becomes 10⁻⁴ with the use of quantum memories, which remains higher to the values reported in [12]. But the point we wish to emphasize concerns the fact that, due to the large success probability of one single stage, our protocol is intended to be all the more advantageous as the target is complex. Furthermore, one has also to keep in mind that the protocol presented here only requires monomode single photon states, no matter which way they were generated (the resource state of Eq. (1) being produced by the protocol of [7] for instance). Recently, some new experimental protocols managed to retrieve on demand high quality single photon Fock states (fidelity of 82%) from cold atomic ensembles [18], which would bring the success probability of the experiment up to 10⁻², showing the interest of our proposal.

The calculations done until now supposed that the photons and the homodyne detection were both perfect, which is obviously not the case in practice: this is what we are going to focus on in the next section.

3.2. Imperfections

Let us now study the influence of imperfections on the protocol. These will be taken of two different types: either from the resource state itself or from the homodyne detection used for the heralding.

For the sake of simplicity, and to picture precisely the effect of the protocol, we will consider in this paragraph the generation of states with a protocol fed by two single photon Fock states (α = 0 in Eq. (1)). The Hong Ou Mandel effect [19] makes the one photon contribution vanish (c₁ = 0 in Eq. (9)), and we are left with superpositions of the kind $|\psi 〉=\frac{1}{\sqrt{1+{\left|c_0^{\prime }\right|}^2}}\left(c_0^{\prime }|0〉+|2〉\right)$ with c′₀ ≥ 0. We will focus on three particular cases: c′₀ = 0 (the two photon Fock state), c′₀ = 1 (equally weighted states) and $c_0^{\prime }=1/\sqrt{2}$ (states created in [13]). The generation of these states can be performed for instance with a symmetrical beamsplitter and a conditioning on $x_1^{\prime }=1/\sqrt{2}$, $x_1^{\prime }=\sqrt{(1+\sqrt{2})/2}$ and x′₁ = 0 respectively. We will not try to optimize the success probability here, as we want to estimate the effects of the imperfections only. The conditioning width Δx will then be taken arbitrarily small.

3.2.1. Imperfections of the single photons

In the case where the photons used to feed the setup are imperfect, the performances of the protocol are deteriorated. The imperfections that we are going to take into account are the most common ones: the photons are no longer pure, but consist in a mixture of the single photon |1〉〈1| with vacuum |0〉〈0|. The respective weights give the quality of the state:

 

Fig. 5 Influence of imperfections of (a) photons and (b) homodyne detections on the quality of the output state for cases c′₀ = 0 (solid blue line), c′₀ = 1 (red dot-dashed line) and $c_0^{\prime }=1/\sqrt{2}$ (dashed green line).

Download Full Size | PDF

An interesting point is that the fidelity does not deteriorate faster as the weight of the two photon Fock state increases in the target state: the state with c′₀ = 1 deteriorates faster than the one with $c_0^{\prime }=1/\sqrt{2}$ for input qualities η_phot > 0.7, even if it has a smaller two photon component.

This figure shows that the quality of the input photons is a key property for the proper realization of the protocol, as the output fidelity is strongly dependent on it. For instance, an input fidelity of 90% of the input single photons leads to an output fidelity of 70% for the two photons Fock state output (c′₀ = 0).

3.2.2. Imperfections of the homodyne detections

Other imperfections that can be taken into account are the detection inefficiencies of the homo-dyne detections used for the conditioning. These imperfections (treated in detail in [13]) can be shown to artificially increase the heralding width, and their effects are shown on Fig. 5(b), for the same three cases mentioned previously.

Their effects are quite different, as the fidelity can remain reasonably high for low detection efficiencies. For instance in the case $c_0^{\prime }=1/\sqrt{2}$, the output fidelity is above 90% for detection efficiencies as low as 55%, revealing that in some cases the protocol can enable the production of high fidelity states, even with poor detection performances.

4. Conclusion

We have proposed a new protocol, which enables the generation of arbitrary superpositions of a given number of photons, by the iterative use of a simple scheme based on a mixing on a beamsplitter followed by a homodyne conditioning measurement. This protocol may offer a significant progress in the quantum engineering of states, as it relies on piece by piece building of the state, and allows for the use of quantum memories in order to improve the success probabilities which can then be very high. Another great advantage of the homodyne conditioning technique is that in some cases it can show a certain robustness against detection inefficiencies. With all the recent advances in quantum memories technologies [16] as well as monomode single photon generation [18, 20], we believe that this proposal will open new perspectives in the field of optical quantum states engineering.