Frequency up-conversion of quantum images

Michael Vasilyev; Prem Kumar

doi:10.1364/OE.20.006644

1. Introduction

Light with quantum correlations among various spatial modes (a quantum image) can be used in quantum communications, computing, and measurements. From a practical point of view, it is convenient to produce such quantum images via optical parametric processes that are pumped by easily available high-power laser sources; e.g., those with wavelengths at 532 nm, 780 nm, 1064 nm, or 1550 nm. The resulting quantum images, however, fall into the infrared region of the electromagnetic spectrum, where arrayed detectors have very poor quantum efficiencies, making it difficult to observe and utilize the spatial quantum features of light. One potential way to overcome these low detection efficiencies is to use phase-sensitive parametric amplifiers, which are uniquely capable of amplifying input signals without degrading their signal-to-noise ratios (noiseless amplification) [1]. This property, in combination with the wide temporal bandwidth of the parametric processes, has already been used to make noiseless amplifiers for fiber-optic communications [2–6]. The wide spatial bandwidth of these processes also enables their use for noiseless amplification of classical images [7–11]. It was also recently proposed to use a phase-sensitive parametric amplifier to noiselessly amplify some quantum images, in which multimode squeezed vacuum is used to suppress the loss of information at the periphery of a soft aperture of a spatially-broadband optical receiver [12]. While this approach allows one to extract the quantum features of the original input light even with low-efficiency detectors (e.g., those available in the infrared region of the electromagnetic spectrum), it can amplify only one quadrature of the input light, making this method unsuitable for quantum images in which the information is encoded in photon-number correlations.

An alternative method for overcoming the low infrared detection efficiency is the sum-frequency-generation (SFG) based up-conversion of images into the visible spectrum [13], where efficient arrayed detectors are readily available. This approach has already been demonstrated to provide high-fidelity up-conversion for the parametric twin-beam state [14], single-photon states [15, 16], and quantum memory [17], all of which were spatially single-mode states. A theory of frequency-degenerate image up-conversion via second-harmonic generation was recently introduced in [18], but its practical implementation requires high input signal powers that restrict the types of quantum states that can be up-converted. Another mechanism of frequency conversion relying on quantum teleportation protocol has also been proposed [19]. A quantum theory for simultaneous parametric amplification and up-conversion in the same crystal has recently been developed [20, 21], but it discussed only the signal-to-noise-ratio evolution of an input coherent-state (i.e., classical) image. In this paper, we discuss the up-conversion of a spatially-multimode quantum state (i.e., a quantum image). In Section 2 we derive the analytical expressions for the up-conversion efficiencies in the case of a plane-wave pump or a short crystal length. In Section 3 we calculate the fidelities of the up-converted images and the pump powers required for up-conversion. We do so by using the example of a quantum image produced via spontaneous parametric down-conversion in a χ⁽²⁾ crystal pumped by a beam carrying orbital angular momentum [22, 23]. Our proposed up-conversion method is not limited only to that particular type of quantum image and can be equally well applied to other types, e.g., to the quantum images with sub-shot-noise spatial correlations [24–26] and continuous-variable entanglement of many spatial modes [27] that have been recently generated. We conclude in Section 4 by summarizing the results.

2. Theory of quantum image conversion

We start with the SFG equations in the undepleted-pump and paraxial approximations with polarized (scalar) fields. Similarly to the derivations of the equations for the inverse process of parametric amplification [28] and using our previous notations [29, 30], we are looking for solutions in the form $e_{i} (\vec{r}, t) = E_{i} (\vec{ρ}, z) e^{i (k_{i} z - ω_{i} t)} + c .c .$ , where $E_{i} (\vec{ρ}, z)$ is the slowly-varying electric-field envelope of the i^th wave propagating in z-direction, $\vec{ρ}$ is a transverse vector with coordinates (x, y), ω_i is the angular frequency, k_i = ω_in_i/c is the wavevector, n_i is the refractive index, and the intensity is given by $I_{i} (\vec{ρ}, z) = 2 ε_{0} n_{i} c | E_{i} (\vec{ρ}, z) |^{2}$ . Subscript i can take values i = 1, 2, 3 so that ω₁ + ω₂ = ω₃. In the presence of a strong pump $E_{2} (\vec{ρ}, z)$ at angular frequency ω₂, the input-signal field $E_{1} (\vec{ρ}, z)$ at angular frequency ω₁ is coupled to the up-converted field $E_{3} (\vec{ρ}, z)$ at angular frequency ω₃ through the following coupled-wave equations:

{\begin{cases} \frac{\partial E_{1} (\vec{ρ}, z)}{\partial z} = \frac{i}{2 k_{1}} \nabla_{ρ}^{2} E_{1} (\vec{ρ}, z) + \frac{2 i ω_{1} d_{eff}}{n_{1} c} E_{2}^{*} (\vec{ρ}, z) E_{3} (\vec{ρ}, z) e^{i Δ k_{FC} z}, \\ \frac{\partial E_{3} (\vec{ρ}, z)}{\partial z} = \frac{i}{2 k_{3}} \nabla_{ρ}^{2} E_{3} (\vec{ρ}, z) + \frac{2 i ω_{3} d_{eff}}{n_{3} c} E_{2} (\vec{ρ}, z) E_{1} (\vec{ρ}, z) e^{- i Δ k_{FC} z}, \end{cases}

where Δk_FC = k₃ – k₁ – k₂ is the wavevector mismatch, d_eff is the effective nonlinear coefficient, and

\nabla_{ρ}^{2} = (\partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2})

. Equation (1) describes the traveling-wave up-converter in the paraxial approximation with a pump of arbitrary spatial profile. Since we are interested in the quantum properties of Eq. (1), it is helpful to use normalized input and up-converted fields a₁ and a₃, respectively, which satisfy commutation relations

[a_{1}, a_{1}^{+}] = δ (\vec{ρ} - {\vec{ρ}}^{'})

,

[a_{3}, a_{3}^{+}] = δ (\vec{ρ} - {\vec{ρ}}^{'})

. Since the normalized fields are related to the original fields by

a_{1, 3} (\vec{ρ}, z) \propto \sqrt{n_{1, 3} / ω_{1, 3}} E_{1, 3} (\vec{ρ}, z)

(where the omitted proportionality constants are not dependent on the frequency of the wave), the nonlinear coupling coefficients of Eq. (1) become symmetric when Eq. (1) is re-written for the normalized fields:

{\begin{cases} \frac{\partial a_{1} (\vec{ρ}, z)}{\partial z} = \frac{i}{2 k_{1}} \nabla_{ρ}^{2} a_{1} (\vec{ρ}, z) + i κ^{*} (\vec{ρ}, z) a_{3} (\vec{ρ}, z) e^{i Δ k_{FC} z}, \\ \frac{\partial a_{3} (\vec{ρ}, z)}{\partial z} = \frac{i}{2 k_{3}} \nabla_{ρ}^{2} a_{3} (\vec{ρ}, z) + i κ (\vec{ρ}, z) a_{1} (\vec{ρ}, z) e^{- i Δ k_{FC} z}, \end{cases}

where

κ (\vec{ρ}, z) = \sqrt{\frac{4 ω_{1} ω_{3} d_{eff}^{2}}{n_{1} n_{3} c^{2}}} E_{2} (\vec{ρ}, z) .

Let us define the spatial-frequency ( $\vec{q}$ ) domain via the direct and inverse Fourier transforms

{\tilde{a}}_{1, 3} (\vec{q}, z) = \int a_{1, 3} (\vec{ρ}, z) e^{- i \vec{q} \cdot \vec{ρ}} d \vec{ρ}, a_{1, 3} (\vec{ρ}, z) = \int {\tilde{a}}_{1, 3} (\vec{q}, z) e^{i \vec{q} \cdot \vec{ρ}} \frac{d \vec{q}}{{(2 π)}^{2}} .

In the absence of the pump (E₂ = 0), Eq. (1) is reduced to the paraxial Helmholtz equation, whose solution for the normalized fields in the Fourier domain is given by

{\tilde{a}}_{1, 3} (\vec{q}, z) = {\tilde{a}}_{1, 3} (\vec{q}, 0) \exp (- i \frac{q^{2}}{2 k_{1, 3}} z),

where

q = | \vec{q} |

. In general, similarly to the case of optical parametric amplification, Eq. (2) has to be solved numerically in order to account for coupling among the various spatial modes due to the effects akin to gain-induced diffraction [31]. Some semi-analytical methods mirroring those used in solving spatially-broadband parametric-amplifier equations [32–36] can be applied here as well. In this paper, however, we will restrict ourselves to the two particular cases that admit analytical solutions: the case of a plane-wave pump and the case of a short crystal with arbitrary pump.

Analytical solution 1: plane-wave pump

With the plane-wave pump $E_{2} (\vec{ρ}, z) = | E_{2} | e^{i θ_{2}} = const .$ , Eq. (1) still retains the shift-invariance of the original paraxial Helmholtz equation and hence it is easily solved in the Fourier domain:

\begin{array}{l} {\tilde{a}}_{1} (\vec{q}, z) = e^{- i \frac{q^{2}}{2 k_{1}} z} [t (q) {\tilde{a}}_{1} (\vec{q}, 0) + r (q) {\tilde{a}}_{3} (\vec{q}, 0)], \\ {\tilde{a}}_{3} (\vec{q}, z) = e^{- i \frac{q^{2}}{2 k_{3}} z} [- r^{*} (q) {\tilde{a}}_{1} (\vec{q}, 0) + t^{*} (q) {\tilde{a}}_{3} (\vec{q}, 0)], \end{array}

where the coefficients of the above unitary input-output transformation are

\begin{array}{l} t (q) = (\cos γ z - \frac{i Δ k_{eff}}{2 γ} \sin γ z) \times \exp (i \frac{Δ k_{eff}}{2} z), \\ r (q) = i \frac{κ^{*}}{γ} \sin γ z \times \exp (i \frac{Δ k_{eff}}{2} z), \end{array}

the effective wavevector mismatch is

Δ k_{eff} = Δ k_{FC} + \frac{q^{2}}{2} (\frac{1}{k_{1}} - \frac{1}{k_{3}}),

and the parametric gain coefficient γ is given by

γ = \sqrt{| κ |^{2} + Δ k_{eff}^{2} / 4} .

Equation (6) couples the input and up-converted fields of the same spatial frequency $\vec{q}$ . This transformation is equivalent to that of a beam-splitter with reflectance (up-conversion efficiency)

\begin{array}{l} R = | r |^{2} \\ = \frac{| κ |^{2}}{γ^{2}} \sin^{2} γ z \end{array}

and transmittance

\begin{array}{l} T = | t |^{2} \\ = 1 - \frac{| κ |^{2}}{γ^{2}} \sin^{2} γ z \\ = 1 - R . \end{array}

The evolution of the quantum field operators also obeys Eq. (6), which leads to quantum correlations between the same spatial-frequency components of the input and the up-converted images. The correlations in the $\vec{ρ}$ -domain, on the other hand, are not ideal, because the limited spatial bandwidth of the up-conversion process, determined by the wavevector mismatch of Eq. (8), results in image distortion. The spatial frequencies, for which the mismatch in (8) is nearly zero, can be up-converted with near-unity efficiency R, whereas those outside the up-conversion spatial bandwidth experience very low conversion efficiency. For input images whose quantum features are contained within the spatial bandwidth of the up-conversion process, the output quantum image will reproduce all the information from the input with high fidelity. And vice versa, for images with broader spatial frequency content, the non-unity conversion efficiency R is equivalent to loss and results in reduction of the fidelity of the up-converted state. We also note that the up-conversion takes place equally for both quadratures of the image, and no underlying symmetry of the image is required (this is in contrast to the phase-sensitive amplification process, which requires symmetry between the signal’s positive- and negative-spatial-frequency components, leading to a flat phase front of the image [7,30,35]).

Analytical solution 2: short crystal with inhomogeneous pump

For sufficiently short crystals, the diffraction terms in Eqs. (1) and (2) can be neglected, and the coupled-wave equations take the following form:

{\begin{cases} \frac{\partial a_{1} (\vec{ρ}, z)}{\partial z} = i κ^{*} (\vec{ρ}) a_{3} (\vec{ρ}, z) e^{i Δ k_{FC} z}, \\ \frac{\partial a_{3} (\vec{ρ}, z)}{\partial z} = i κ (\vec{ρ}) a_{1} (\vec{ρ}, z) e^{- i Δ k_{FC} z} . \end{cases}

Here κ, defined in (3), is, in general, a complex parameter that depends on the coordinate $\vec{ρ}$ (if the pump is inhomogeneous) and, at the same time, due to the short crystal length, the z-dependence of the pump is neglected. One can then introduce parameters t and r as

\begin{array}{l} t (\vec{ρ}, z) = (\cos γ z - \frac{i Δ k_{FC}}{2 γ} \sin γ z) \times \exp (i \frac{Δ k_{FC}}{2} z), \\ r (\vec{ρ}, z) = i \frac{κ^{*}}{γ} \sin γ z \times \exp (i \frac{Δ k_{FC}}{2} z), \end{array}

where

γ (\vec{ρ}) = \sqrt{| κ |^{2} (\vec{ρ}) - Δ k_{FC}^{2} / 4}, | κ |^{2} (\vec{ρ}) = \frac{2 ω_{1} ω_{3} d_{eff}^{2} I_{2} (\vec{ρ})}{ε_{0} n_{1} n_{3} n_{2} c^{3}},

so that the solution takes the form of point-by-point (pixel-by-pixel) up-conversion with efficiency R = |r|²:

\begin{array}{l} a_{1} (\vec{ρ}, z) = t (\vec{ρ}, z) a_{1} (\vec{ρ}, 0) + r (\vec{ρ}, z) a_{3} (\vec{ρ}, 0), \\ a_{3} (\vec{ρ}, z) = - r^{*} (\vec{ρ}, z) a_{1} (\vec{ρ}, 0) + t^{*} (\vec{ρ}, z) a_{3} (\vec{ρ}, 0), \end{array}

which is equivalent to the beam-splitter transformation.

3. Example of quantum image conversion

In this section, we will provide an example of frequency conversion of the quantum image [22, 23, 37] produced via the frequency-degenerate spontaneous parametric down-conversion (SPDC) process of type-I with spatially-modulated z-propagating pump E_p(x, y) of frequency ω_p = 2ω₁ (here we use subscript “p” to distinguish the SPDC pump from the up-conversion pump denoted by subscript “2”). Assuming that the SPDC crystal length l_SPDC is much smaller than the Rayleigh distance of the pump, the SPDC state can be approximated for low pump intensities (i.e., neglecting multiple-pair generation) by an unnormalized wavefunction

| ψ 〉 = | 0 〉 + \int d {\vec{q}}_{s} d {\vec{q}}_{i} F ({\vec{q}}_{s}, {\vec{q}}_{i}) {| 1 〉}_{{\vec{q}}_{s}} {| 1 〉}_{{\vec{q}}_{i}},

where

{\vec{q}}_{s}

and

{\vec{q}}_{i}

are the transverse components (spatial frequencies) of the signal and idler wavevectors

{\vec{k}}_{s}

and

{\vec{k}}_{i}

, respectively, and

{| 1 〉}_{{\vec{q}}_{s}} {| 1 〉}_{{\vec{q}}_{i}}

denotes a state having one photon with spatial frequency

{\vec{q}}_{s}

and one photon with spatial frequency

{\vec{q}}_{i}

, with all other modes in the vacuum state. F is the joint two-photon wavefunction in the momentum space given by

\begin{array}{l} F ({\vec{q}}_{s}, {\vec{q}}_{i}) \propto \int d \vec{r} E_{p} (\vec{r}) e^{i Δ \vec{k} \cdot \vec{r}} \\ = \int d z e^{i Δ k_{z} z} \int d x d y E_{p} (x, y) e^{i (Δ k_{x} x + Δ k_{y} y)} \\ = l_{SPDC} e^{i Δ k_{z} l_{SPDC} / 2} sinc (\frac{Δ k_{z} l_{SPDC}}{2}) {\tilde{E}}_{p} ({\vec{q}}_{s} + {\vec{q}}_{i}), \end{array}

where the z-integral gives the phase-mismatch factor, and the (x,y)-integral gives the spatial Fourier transform

{\tilde{E}}_{p} (\vec{q})

of the transverse spatial profile of the pump, with

\begin{array}{l} Δ \vec{k} = {\vec{k}}_{p} - {\vec{k}}_{s} - {\vec{k}}_{i} = \hat{z} Δ k_{z} - {\vec{q}}_{s} - {\vec{q}}_{i} \\ \approx \hat{z} (k_{p} - k_{s} - k_{i} + \frac{q_{s}^{2}}{2 k_{s}} + \frac{q_{i}^{2}}{2 k_{i}}) - {\vec{q}}_{s} - {\vec{q}}_{i} \\ \approx \hat{z} \frac{q_{s}^{2} + q_{i}^{2} - 2 q_{0}^{2}}{2 k_{s}} - {\vec{q}}_{s} - {\vec{q}}_{i}, \end{array}

q_{s, i}^{2} = {(k_{x})}_{s, i}^{2} + {(k_{y})}_{s, i}^{2}

and

q_{0}^{2} = - k_{s} (k_{p} - 2 k_{s})

. In the last line of Eq. (18) we assumed k_i ≈k_s.

The peculiar properties of the state of Eq. (16), obtained with orbital-momentum-carrying pump, are illustrated in the far-field (transverse momentum space) diagram shown in Fig. 1(a) , where pump profile of Laguerre-Gaussian mode LG₀₂ with orbital momentum l = 2 is shown in purple as an example. For non-collinear phase matching (i.e., for k_p ≠ 2k_s), the SPDC photons are emitted in the pattern of a phase-matched cone [red ring of radius q₀ in Fig. 1(a)]. The image obtained by direct detecting the SPDC simply reflects this ring shape [see insert 1 in Fig. 1(a)]. The coincidence count measurement, on the other hand, reveals the underlying quantum structure of this image. In particular, if an idler photon at spatial frequency ${\vec{q}}_{i 0}$ is detected (heralded) by a small-area detector [yellow dot in Fig. 1(a)], then the probability amplitude of finding a signal photon with spatial frequency ${\vec{q}}_{s}$ (or, in other words, the spatial profile of the resulting heralded single-photon mode) will be proportional to the function $F ({\vec{q}}_{s}, {\vec{q}}_{i 0})$ from Eq. (17). The magnitude of this function is a product of the spatial profile of the phase-matching condition [red ring in Fig. 1(a)] and the far-field pattern of the SPDC pump centered at $- {\vec{q}}_{i 0}$ [checkered purple shape in Fig. 1(a), which has the profile of a ring if the pump field is given by LG₀₂]. The resulting product is sketched by the two green dots in Fig. 1(a), while its actual calculated shape is shown by the insert 3 in the same Figure. If the pump is a simple fundamental Gaussian beam LG₀₀, this product takes the form shown by the insert 2. Thus, the spatial mode of the heralded single-photon state of the signal depends on the pump profile. This property is only observable in coincidence counting and represents a quantum image.

Fig. 1 (a) Illustration of the phase matching, as well as the quantum image generation and up-conversion in the far field (transverse momentum space) when both the SPDC pump and the up-conversion pump are propagating in the same direction. (b) A schematic of the corresponding experimental setup. Inserts 2 and 3 in (a) show the quantum image function $| F ({\vec{q}}_{s}, {\vec{q}}_{i 0}) |^{2}$ for the cases of SPDC pumping by LG₀₀ and LG₀₂ modes, respectively, and insert 1 is the image obtained by scanning a single photon-counting detector across the output plane in the far field. All three inserts are equally applicable to the configuration shown in Fig. 2. SPDC–spontaneous parametric down-conversion, FC–frequency up-conversion.

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Next, we study the up-conversion of the SPDC state of Eq. (16). Since the underlying quantum image is revealed in coincidence counting, it is convenient to use the same heralding procedure as that described above to check the fidelity of the up-converted quantum image (we use this procedure for convenience of calculation only; heralding is not required for up-conversion scheme itself to work). In order to up-convert the quantum image given by Eq. (16), we will assume that the output of the SPDC crystal is imaged into the input of the up-conversion crystal of length l_FC by a 1:1 telescope and that the up-conversion pump is a plane wave (i.e., the conversion obeys the formulas for Analytical Solution 1 in Section 2), propagating in the same direction as the SPDC pump, as depicted in Fig. 1(b). The mode at frequency ω₃ is in the vacuum state at the input of the up-converter. For a single-photon input at frequency ω₁ and spatial frequency ${\vec{q}}_{s}$ or ${\vec{q}}_{i}$ , the input-output transformation (6) performs a beam-splitting procedure on signal and idler modes independently, which takes the following form in Schrödinger picture:

{| Ψ 〉}_{{\vec{q}}_{s, i}} = t^{*} ({\vec{q}}_{s, i}) {| 10 〉}_{{\vec{q}}_{s, i}} - r^{*} ({\vec{q}}_{s, i}) {| 01 〉}_{{\vec{q}}_{s, i}},

wherein the transformation coefficients

t ({\vec{q}}_{s, i}), r ({\vec{q}}_{s, i})

are given by Eq. (7) with z = l_FC, and

{| 10 〉}_{\vec{q}}

(or

{| 01 〉}_{\vec{q}}

) refers to a state having one photon with spatial frequency

\vec{q}

and frequency ω₁ (or ω₃, respectively), with all other modes in a vacuum state. Thus, the SPDC state (16) after up-conversion becomes:

\begin{array}{l} {| ψ 〉}_{FC} = | 0 〉 + \int d {\vec{q}}_{s} d {\vec{q}}_{i} F ({\vec{q}}_{s}, {\vec{q}}_{i}) t^{*} ({\vec{q}}_{s}) t^{*} ({\vec{q}}_{i}) {| 10 〉}_{{\vec{q}}_{s}} {| 10 〉}_{{\vec{q}}_{i}} \\ + \int d {\vec{q}}_{s} d {\vec{q}}_{i} F ({\vec{q}}_{s}, {\vec{q}}_{i}) r^{*} ({\vec{q}}_{s}) r^{*} ({\vec{q}}_{i}) {| 01 〉}_{{\vec{q}}_{s}} {| 01 〉}_{{\vec{q}}_{i}} \\ - \int d {\vec{q}}_{s} d {\vec{q}}_{i} F ({\vec{q}}_{s}, {\vec{q}}_{i}) r^{*} ({\vec{q}}_{s}) t^{*} ({\vec{q}}_{i}) {| 01 〉}_{{\vec{q}}_{s}} {| 10 〉}_{{\vec{q}}_{i}} \\ - \int d {\vec{q}}_{s} d {\vec{q}}_{i} F ({\vec{q}}_{s}, {\vec{q}}_{i}) t^{*} ({\vec{q}}_{s}) r^{*} ({\vec{q}}_{i}) {| 10 〉}_{{\vec{q}}_{s}} {| 01 〉}_{{\vec{q}}_{i}} . \end{array}

In Eqs. (19) and (20) we have omitted the non-essential diffractive phase q²z/(2k₃), which is equivalent to observing the far-field pattern in the Fourier plane of a lens.

Upon detection of the idler photon with transverse wavevector ${\vec{q}}_{i 0}$ , the up-converted state can be reduced to a heralded up-converted state

{| ψ 〉}_{Heralded, FC} = \int d {\vec{q}}_{s} F ({\vec{q}}_{s}, {\vec{q}}_{i 0}) [t^{*} ({\vec{q}}_{s}) {| 10 〉}_{{\vec{q}}_{s}} - r^{*} ({\vec{q}}_{s}) {| 01 〉}_{{\vec{q}}_{s}}] .

The fidelity η of the frequency up-converted quantum image can be quantified by the overlap of the heralded states obtained with ideal and non-ideal frequency conversion (the former corresponding to Eq. (21) with r = 1 and t = 0 and denoted simply “Heralded” below; it is also the same as the ideal up-conversion of the state heralded prior to frequency conversion):

η = \frac{{|_{Heralded} {〈 ψ | ψ 〉}_{Heralded, ​ FC} |}^{2}}{_{Heralded} {〈 ψ | ψ 〉}_{Heralded}_{Heralded, FC} {〈 ψ | ψ 〉}_{Heralded, ​ FC}} = \frac{{| \int d {\vec{q}}_{s} | F ({\vec{q}}_{s}, {\vec{q}}_{i 0}) |^{2} r^{*} ({\vec{q}}_{s}) |}^{2}}{{[\int d {\vec{q}}_{s} | F ({\vec{q}}_{s}, {\vec{q}}_{i 0}) |^{2}]}^{2}},

which is, in turn, determined by the overlap of the two-photon wavefunction F and the up-conversion coefficient r* in the transverse momentum space. Thus, to maximize the fidelity of the up-converted quantum image, one needs to ensure that r* is close to unity over the range of spatial frequencies where F has non-zero values. We note that the fidelity is the same regardless of whether the idler photon is detected before or after up-conversion (however, the probability of detection of the idler photon will be lower after imperfect up-conversion).

To achieve appreciable count rates, it makes sense to place the idler detector in the vicinity of q_i₀ = q₀ (which corresponds to Δk_z = 0 in the plane-wave pump case), so we will assume ${\vec{q}}_{i 0} = - \hat{x} q_{0}$ . Then, the up-conversion should be set up so that Δk_eff = 0 takes place at q = q_s = q₀, i.e.,

\begin{array}{l} Δ k_{FC} = - \frac{q_{0}^{2}}{2} (\frac{1}{k_{1}} - \frac{1}{k_{3}}) < 0, \\ Δ k_{eff} = \frac{q_{s}^{2} - q_{0}^{2}}{2} (\frac{1}{k_{1}} - \frac{1}{k_{3}}), \end{array}

and k₁=k_s. Thus, the phase-matched bands of both the up-conversion and the SPDC are centered at

| {\vec{q}}_{s} | = q_{0}

, and to achieve the high fidelity we just need to ensure that the spatial bandwidth of the up-conversion [represented by the half-width of the blue ring in Fig. 1(a)] is greater than that of the SPDC [represented by the half-width of the red ring in Fig. 1(a)]. To estimate the former, we note that at

| {\vec{q}}_{s} | = q_{0}

, r* = 1 for γl_FC = |κ|l_FC = π/2, i.e. l_FC = π/(2|κ|). The first zero of r* in Eq. (7) takes place at

| {\vec{q}}_{s} | = q_{0} + Δ q_{c}^{FC}

so that γl_FC = π (i.e.,

Δ k_{eff} = π \sqrt{3} / l_{F C}

), where the detuning from the center to the first zero quantifies the spatial bandwidth of the up-conversion:

Δ q_{c}^{FC} = \frac{π \sqrt{3}}{q_{0} l_{FC}} \frac{k_{1} k_{3}}{k_{3} - k_{1}} .

Similarly, the spatial bandwidth of the SPDC $Δ q_{c}^{SPDC}$ can be estimated from the first zero of the sinc function in Eq. (17) by using Δk_z from Eq. (18) with q_i = q₀ and $| {\vec{q}}_{s} | = q_{0} + Δ q_{c}^{SPDC}$ :

\begin{array}{l} Δ q_{c}^{SPDC} = \frac{2 π k_{s}}{q_{0} l_{SPDC}} \\ = Δ q_{c}^{FC} \frac{2}{\sqrt{3}} \frac{l_{FC} n_{s}}{l_{SPDC} n_{1}} (1 - \frac{k_{1}}{k_{3}}) \\ = Δ q_{c}^{FC} \frac{2 ω_{2} l_{FC} n_{s}}{\sqrt{3} ω_{3} l_{SPDC} n_{1}} . \end{array}

High fidelity is realized when $Δ q_{c}^{FC} > Δ q_{c}^{SPDC}$ , i.e.,

\frac{l_{SPDC}}{l_{FC}} > \frac{2 ω_{2} n_{s}}{\sqrt{3} ω_{3} n_{1}},

where n_s / n₁ is the ratio of the signal-wave refractive indices in the SPDC and up-conversion crystals.

It is worth noting that one can also try an alternative configuration of the up-conversion pumping, where the up-conversion pump wavevector is collinear with the ${\vec{k}}_{i 0} - 2 {\vec{q}}_{i 0}$ direction, which corresponds to the direction of the signal SPDC photon in the plane-wave-pump approximation. The corresponding phase-matching diagram in the far field and the up-conversion setup are shown in Figs. 2(a) and 2(b), respectively. The SPDC state is still the same as that in Fig. 1, but at the up-converted stage $r^{*} ({\vec{q}}_{s})$ in Eq. (22) should be replaced by $r^{*} (\vec{q})$ , where $\vec{q} = {\vec{q}}_{s} + {\vec{q}}_{i 0}$ . Equation (23) should be replaced by

Fig. 2 (a) Illustration of the phase matching, as well as the quantum image generation and up-conversion in the far field (transverse momentum space) when the up-conversion pump is propagating along the ${\vec{k}}_{i 0} - 2 {\vec{q}}_{i 0}$ direction. (b) A schematic of the corresponding experimental setup. For SPDC pumping by LG₀₂, the quantum image function $| F ({\vec{q}}_{s}, {\vec{q}}_{i 0}) |^{2}$ is proportional to the product of the red and checkered-purple rings in (a), which is sketched as two green dots. The single- and coincidence-count images for SPDC pumping by LG₀₀ and LG₀₂ modes are still represented by inserts 1–3 in Fig. 1. SPDC–spontaneous parametric down-conversion, FC–frequency up-conversion.

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\begin{array}{l} Δ k_{FC} = 0, \\ Δ k_{eff} = \frac{q^{2}}{2} (\frac{1}{k_{1}} - \frac{1}{k_{3}}) . \end{array}

Then, the spatial bandwidth of the up-conversion in this configuration [radius of the blue circle in Fig. 2(a)] is still obtained from the condition $Δ k_{eff} = π \sqrt{3} / l_{FC}$ and is given by

\begin{array}{l} {(Δ q_{c}^{FC})}^{'} = \sqrt{\frac{2 \sqrt{3} π}{l_{FC}} \frac{k_{1} k_{3}}{(k_{3} - k_{1})}} \\ = Δ q_{c}^{FC} \sqrt{\frac{2 q_{0}}{Δ q_{c}^{FC}}} > > Δ q_{c}^{FC}, \end{array}

i.e., the condition

(Δ q_{c}^{FC})^{'} > Δ q_{c}^{SPDC}

is easier to satisfy (it requires lower up-conversion pump power) than the condition

Δ q_{c}^{FC} > Δ q_{c}^{SPDC}

in the previous configuration. However, the alternative configuration is subject to an additional condition that

(Δ q_{c}^{FC})^{'} > Δ q_{p}

, where Δq_p is the spatial bandwidth of the SPDC pump beam [i.e., the blue circle in Fig. 2(a) should be larger than the checkered purple ring]. For the quantum images of interest,

Δ q_{p} > Δ q_{c}^{SPDC}

and, therefore, the requirement on the up-conversion spatial bandwidth to be greater than the pump spatial bandwidth becomes dominant for high fidelity, and may not be easy to satisfy. Note that in the original configuration of Fig. 1 (where the SPDC and up-conversion pumps are collinear) the pump spatial bandwidth does not significantly impact the fidelity as long as Eq. (26) is satisfied.

In the calculations of the spatial bandwidth and the fidelity for the configurations of both Fig. 1 (up-conversion pump co-propagating with the SPDC pump) and Fig. 2 (up-conversion pump propagating in the direction of the SPDC signal), two dimensionless parameters are of great importance. The first is the ratio of the center spatial frequency q₀ to the SPDC spatial bandwidth $Δ q_{c}^{SPDC}$ from Eq. (25):

N = \frac{q_{0}}{Δ q_{c}^{SPDC}} = \frac{q_{0}^{2} l_{SPDC}}{2 π k_{s}} .

The second is related to the square root of the ratio of the SPDC pump’s Rayleigh distance z_R = k_pa₀_p² (where a₀_p is the 1/e intensity radius at the pump’s waist) to l_SPDC:

M = \sqrt{\frac{2 π z_{R}}{l_{SPDC}}} .

The SPDC spatial bandwidth given by Eq. (25) is the spatial bandwidth in the so-called “bandpass” configuration of the optical parametric process, where the signal and idler beams are centered at non-zero spatial frequency q₀. It is also helpful to calculate the spatial bandwidth (at the first zero of the sinc function) of the signal for the “lowpass” configuration (i.e., assuming q_i = q₀ = 0):

Δ q_{c}^{LP-SPDC} = \sqrt{\frac{4 π k_{s}}{l_{SPDC}}},

which allows us to express the dimensionless parameters M and N as

\begin{array}{l} M = a_{0 p} Δ q_{c}^{LP-SPDC}, \\ N = 2 {(\frac{q_{0}}{Δ q_{c}^{LP-SPDC}})}^{2}, \end{array}

and

a_{0 p} q_{0} = M \sqrt{\frac{N}{2}} .

The dependence of the up-conversion fidelity upon the pump’s orbital angular momentum l, the parameter M, and the ratio of the up-conversion to the SPDC crystal lengths l_FC / l_SPDC is shown in Fig. 3 for ω₂ / ω₃ = 2/3 and N = 10. The results indicate that high fidelity can be achieved in all cases by having a sufficiently short l_FC. However, because shortening l_FC requires increasing the up-conversion pump power, the plots show that the configuration of Fig. 2 is clearly preferable as it achieves higher fidelity for a given up-conversion crystal length.

Fig. 3 The fidelity of the up-converted quantum image as a function of the ratio between the lengths of the up-conversion and SPDC crystals. Black dash-dotted traces correspond to the configuration of Fig. 1, whereas colored solid and dashed traces correspond to the configuration of Fig. 2. The cases of orbital momenta l = 0, l = 2, and l = 4 correspond to the SPDC pumping by LG₀₀, LG₀₂, and LG₀₄ modes, respectively. We do not show the black traces with M = 3, because they are almost indistinguishable from the black traces with M = 4.

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Let us consider a specific case of using λ₂ = λ_p = 780 nm, λ₁ = 2λ₂ = 1560 nm, and the up-converted wavelength λ₃ = (1/λ₁ + 1/λ₂)^–1 = 2λ₂/3 = λ₁/3 = 520 nm. Both SPDC and wavelength conversion can be implemented, for example, in PPLN crystals (n_s = n₁ ≈n₂ = n_p ≈n₃ ≈2.14). The (one-sided) angle at which the signal beam emerges in the free space is θ_s = q₀n_s / k_s. Assuming the parameters θ_s = 0.04 rad, M = 4, and N = 10 — which are similar to those in [23] (θ_s = 0.07 rad, M = 4, and N = 8.3), but at a different signal wavelength — we obtain l_SPDC = N λ_s n_s / θ_s² = 20.9 mm and a₀_p = (λ_p l_SPDC / n_p)^1/2M/(2π) = 55.5 μm. Then, in order to achieve the fidelity η = 99% with the configuration of Fig. 2, one needs to employ the up-conversion crystals with lengths l_FC = 16.7 mm, 6.26 mm, and 4.17 mm for the cases of SPDC pumping by the LG₀₀, LG₀₂, and LG₀₄ modes, respectively. This corresponds to the peak up-conversion pump intensities of 2.4, 17.2, and 38.7 MW/cm², respectively, where all three intensity numbers are well below the ~250-MW/cm² optical damage threshold of the lithium-niobate crystal. In contrast to the case of Fig. 2, the configuration of Fig. 1 requires significantly shorter lengths l_FC of 2.5 mm, 1.67 mm, and 1.46 mm to achieve the same fidelity, taking the up-conversion pump requirements to the intensity levels of 108, 241, and 318 MW/cm², respectively, where only the first number is safely below the crystal damage threshold. While the configuration of Fig. 2 requires lower pump intensities, it also makes it more difficult to spatially separate the up-converted image from the up-conversion pump. For our chosen example, where the up-conversion pump’s near-infrared wavelength lies within the spectral response of silicon detectors, a strong dichroic filtering might be needed to prevent the pump beam from affecting the signal detection.

4. Summary

We have developed the theory of frequency up-conversion of an arbitrary spatially-broadband quantum state (quantum image), with analytical solutions derived for the cases of a) plane-wave pump and b) short crystal with arbitrary pump. Using an example of the quantum image imposed by the orbital angular momentum of the pump beam in spontaneous parametric down-conversion, we have obtained the up-conversion fidelities in two up-converter configurations: one where the SPDC and up-conversion pumps co-propagate in the respective crystals and the other where the up-conversion pump co-propagates with the signal beam in the up-conversion crystal. We have shown that the former configuration requires unreasonably high levels of pump intensity, whereas in the latter configuration, 99%-fidelity quantum image up-conversion from 1560 nm to 520 nm wavelength can be obtained in PPLN crystals at moderate up-conversion pump intensities well below the PPLN damage threshold. This indicates the practicality of the quantum image up-conversion with readily available solid-state or fiber-based pump sources.

Acknowledgments

This work was supported in part by DARPA’s Quantum Sensor Program under AFRL Contract No. FA8750-09-C-0194. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA or the U.S. Air Force.

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Frequency up-conversion of quantum images

Abstract

1. Introduction

2. Theory of quantum image conversion

Analytical solution 1: plane-wave pump

Analytical solution 2: short crystal with inhomogeneous pump

3. Example of quantum image conversion

4. Summary

Acknowledgments

References and links

Cited By

Figures (3)

Equations (33)

Optics Express