---
title: Photon-Subtracted Two-Mode Squeezed Fock States
url: https://www.emergentmind.com/topics/photon-subtracted-two-mode-squeezed-fock-state-ps-tmsfs
type: topic
---

# Photon-Subtracted Two-Mode Squeezed Fock States

Photon-subtracted two-mode squeezed Fock states (PS-TMSFSs) are non-Gaussian conditional states obtained by combining two-mode squeezing, ancillary vacuum modes, beam-splitter taps, and photon-number-resolving heralding. In the most explicit two-mode formulation, one begins with a two-mode squeezed Fock input $S_2(r)\lvert m_1,m_2\rangle$ and conditions on subtracting $n_1$ and $n_2$ photons from the two modes; in adjacent measurement-based formulations, the same operational motif is used to herald single-mode squeezed Fock outputs from a two-mode entangled Gaussian resource rather than to retain a two-mode entangled output [2606.25471] [2312.14643]. The subject therefore sits at the intersection of non-Gaussian state engineering, continuous-variable entanglement manipulation, and conditional photonic measurement.

## 1. Definition and terminological scope

A two-mode squeezed Fock state is generated by acting with the two-mode squeezing operator
$$
S_{2}(r)=\exp\bigl[r\,(a_{1}^{\dagger}a_{2}^{\dagger}-a_{1}a_{2})\bigr]
$$
on a two-mode Fock input,
$$
\lvert\psi_{\rm TMSFS}\rangle=S_{2}(r)\,\lvert m_{1},m_{2}\rangle_{A_{1}A_{2}}.
$$
Photon subtraction is then implemented by mixing each mode with vacuum on ancillary beam splitters and conditioning on photon-number-resolving detections in the tapped ports; successful clicks $(n_1,n_2)$ define the photon-subtracted state [2606.25471].

The label PS-TMSFS appears in adjacent but non-identical constructions. Some works use it for an entangled two-mode non-Gaussian resource obtained after subtracting photons from a genuine $S_2(r)\lvert m_1,m_2\rangle$ input. Other works use closely related machinery to generate squeezed Fock states by measurement from a generic two-mode entangled Gaussian state. A further strand uses a two-mode squeezed vacuum plus weak mode mixing and $n$-photon heralding to prepare a small-dimensional “core” state that is later converted into a squeezed coherent-state superposition [1503.08970] [2407.11193].

A notable boundary case is provided by the general theory of photon subtraction for entangled beams based on two-mode $SU(2)$ coherent states. That framework supplies a compact conditional-output formalism, but it does not itself introduce a two-mode squeezed Fock state $\lvert r;n,m\rangle=S_2(r)\lvert n,m\rangle$ as its working input family; its explicit formulas instead assume a single-mode superposition entering one port of a $50{:}50$ beam splitter and vacuum in the other [2108.12528].

| Work | State or protocol | Relation to PS-TMSFS |
|---|---|---|
| [2606.25471] | $S_2(r)\lvert m_1,m_2\rangle$ plus conditional subtraction $(n_1,n_2)$ | Direct two-mode PS-TMSFS resource |
| [2312.14643], [2407.11193] | Two-mode entangled Gaussian resource plus number projection | Measurement-based squeezed-Fock generation |
| [1503.08970] | TMSV, weak beam-splitter mixing, $n$-photon heralding | PS-TMSFS “core” for cat-state synthesis |
| [2108.12528] | General subtraction theory for entangled beams | Formal backdrop, not a direct TMSFS construction |

## 2. Conditional generation protocols

In the direct two-mode construction, the input $\lvert m_1,m_2\rangle_{A_1A_2}$ is first transformed by $S_2(r)$ and then coupled to ancillary vacua $F_1,F_2$ through beam splitters
$$
B(T)=\exp\bigl[\theta(\,a_{A}^{\dagger}a_{F}-a_{A}a_{F}^{\dagger})\bigr], \qquad T=\cos^{2}\theta.
$$
Photon-number-resolving detection projects the ancillas onto $\lvert n_1\rangle\langle n_1|\otimes \lvert n_2\rangle\langle n_2|$, and successful heralds subtract $n_1$ and $n_2$ photons from the two system modes. The normalized output is obtained by dividing the unnormalized characteristic function by the success probability
$$
P^{\rm PS}
=\widetilde\chi_{A_{1}'A_{2}'}^{\rm PS}(0)
=\bigl[\hat F\,a_{0}\,e^{u^{T}M_{3}u}\bigr]_{u=0},
$$
where $\hat F$ is the differential operator associated with the Fock inputs and subtraction pattern, and $M_{1,2,3}$ are the real matrices tabulated for the protocol [2606.25471].

The vacuum-seeded sector, $\lvert m_1,m_2\rangle=\lvert0,0\rangle$, reduces to the two-mode squeezed vacuum (TMSV),
$$
\lvert\Psi_{\rm TMSV}(\lambda)\rangle
=\sqrt{1-\lambda^2}\sum_{n=0}^{\infty}\lambda^n\lvert n,n\rangle,
\qquad \lambda=\tanh |\xi|.
$$
Simultaneous subtraction of $k$ photons from both modes produces the unnormalized state
$$
\lvert\Psi^{(k)}\rangle_{\rm un}
=\sqrt{1-\lambda^2}\sum_{j=0}^{\infty}
\lambda^{j+k}\frac{(j+k)!}{j!}\,\lvert j,j\rangle,
$$
up to the normalization factor $\mathcal N_k$ [1901.00122]. This sector is important because it is both analytically tractable and experimentally established.

A small-mixing approximation leads to a particularly compact heralded state. After weak beam-splitter mixing of TMSV signal and idler, $n$-photon detection in the heralding arm produces
$$
\lvert\Psi_n\rangle_{s'}=
N_n\bigl[\epsilon\sqrt{n(n-1)}\,\lvert n-2\rangle+\lambda\,\lvert n\rangle\bigr],
\qquad
N_n=\frac{1}{\sqrt{n(n-1)\epsilon^2+\lambda^2}},
$$
with approximate success probability
$$
P_n(\lambda)\approx (1-\lambda^2)\lambda^{2(n-2)}
\bigl[n(n-1)\epsilon^2+\lambda^2\bigr].
$$
This two-component non-Gaussian “core” is the state later converted into a squeezed even coherent-state superposition [1503.08970].

## 3. Representations in Fock space, quadrature space, and phase space

The PS-TMSFS admits a natural phase-space description in terms of a Gaussian kernel dressed by derivatives. In the two-mode formulation, the unnormalized output characteristic function is
$$
\widetilde\chi_{A_{1}'A_{2}'}^{\rm PS}(\Lambda)
=\hat F\,a_{0}\,
\exp\!\bigl[\Lambda^{T}M_{1}\Lambda
+u^{T}M_{2}\Lambda+u^{T}M_{3}u\bigr],
$$
and the normalized characteristic function is
$$
\chi_{A_{1}'A_{2}'}^{\rm PS}(\Lambda)
=\frac{\widetilde\chi_{A_{1}'A_{2}'}^{\rm PS}(\Lambda)}{P^{\rm PS}}.
$$
Because the differential operator $\hat F$ acts on the Gaussian kernel, the output is not merely Gaussian: it carries a polynomial prefactor in the phase-space variables, with highest total degree $2(m_1+m_2+n_1+n_2)$ whenever subtraction is present [2606.25471]. This is the direct phase-space signature of non-Gaussianity.

An equivalent quadrature-space representation is available for measurement-generated squeezed-Fock states derived from a generic two-mode Gaussian wavefunction with parameters $a,b,d$. After number projection on tapped ancillas, the heralded wavefunction assumes the form
$$
\Psi_{m,n}(x_A,x_B)
=\mathcal{N}_{m,n}^{-1/2}(A,B,D)\,
H_m(u_A)\,H_n(u_B)\,
\exp\!\bigl[-\tfrac12(e^{2R_A}x_A^2+e^{2R_B}x_B^2)\bigr],
$$
where $H_m$ and $H_n$ are Hermite polynomials, and the effective squeezings $R_A,R_B$ are fixed by the universal-solution conditions [2407.11193]. The Hermite structure makes explicit that photon subtraction promotes the Gaussian core into a polynomial-times-Gaussian state, which is precisely the squeezed-Fock pattern.

For the vacuum-seeded case, the Wigner function can be generated by applying differential operators to the known TMSV Wigner function,
$$
W^{(k)}(\alpha,\beta)\propto
(\partial_\alpha\,\partial_\beta)^k\,W_{\rm TMSV}(\alpha,\beta),
$$
so that each subtraction order introduces additional oscillations, negative lobes, and interference fringes in phase space [1901.00122]. This suggests a common structural picture across formulations: subtraction converts a Gaussian entangled resource into a non-Gaussian state whose analytic expressions remain controlled by derivatives of Gaussian objects.

## 4. Universal-solution regime and optical implementations

A central analytical result in the measurement-based literature is the “universal-solution” regime, in which the heralded output exactly matches the ideal $n$th squeezed-Fock wavefunction for arbitrary $n$. For a non-displaced two-mode entangled Gaussian state with wavefunction parameters $a,b,d$, exact equality holds if and only if
$$
d-\frac{b^2}{a+1}=e^{2r}, \qquad
\frac{b^2}{a^2-1}=e^{2r}, \qquad
\Im\!\Bigl(d-\frac{b^2}{a+1}\Bigr)=0.
$$
Eliminating $r$ gives the real constraint
$$
\Re\!\Bigl(d-\frac{b^2}{a+1}\Bigr)
=
\frac{b^2}{a^2-1}.
$$
In that regime, the heralded state coincides with $\Psi_{\rm SF}(x;n,r)$ exactly for any $n$ [2312.14643].

The case $n=1$ is exceptional. Because $H_1(x)=2x$ is linear, one-photon subtraction from an arbitrary non-displaced two-mode entangled Gaussian state always yields a first squeezed-Fock state $S(r)\lvert1\rangle$, with
$$
e^{2r}=\Re\!\Bigl(d-\frac{b^2}{a+1}\Bigr).
$$
The corresponding heralding probability satisfies $P_{1,\max}=1/4$, and that maximum is attained precisely in the universal-solution regime [2312.14643]. This is a sharp statement about the privileged status of first-order subtraction.

Two optical implementations are treated in detail. In the beam-splitter (BS) approach, two single-mode squeezed vacua of parameters $r_1,r_2$ are mixed with power transmission $t$, producing
$$
a=e^{2r_1}(1-t)+e^{2r_2}t,\quad
b=\sqrt{t(1-t)}(e^{2r_1}-e^{2r_2}),\quad
d=e^{2r_1}t+e^{2r_2}(1-t).
$$
In the controlled-$Z$ (CZ) approach,
$$
a=e^{2r_1},\qquad b=i\,g,\qquad d=e^{2r_2}.
$$
The BS protocol has energy cost
$$
E_{BS}=\sinh^2(r_1)+\sinh^2(r_2),
$$
whereas the CZ protocol requires
$$
E_{CZ}=\sinh^2(r_1)+\sinh^2(r_2)+2\,\sinh^2(r_{an}),
$$
because it uses two auxiliary squeezed ancilla modes. The ideal success probability peaks at approximately $25\%$ for $(m,n)=(1,0)$ or $(0,1)$; BS requires only vacuum taps, while CZ requires two ancilla squeezers plus passive optics and becomes competitive only if large-$g$ CZ gates and high ancilla squeezing are available [2407.11193].

## 5. Nonclassicality, statistics, and entanglement

For the vacuum-seeded PS-TMSFS sector, subtraction changes both local and joint photon statistics in a highly structured way. The single-mode second-order correlation function
$$
g_j^{(2)}=
\frac{\langle \hat a_j^\dagger\hat a_j^\dagger\hat a_j\hat a_j\rangle}
{\langle \hat a_j^\dagger\hat a_j\rangle^2}
$$
tends toward $1$ with increasing subtraction order $k$, indicating that each marginal becomes closer to Poissonian [1901.00122]. Experimentally, the marginals evolve from thermal at $k=0$ to nearly Poisson distributions of mean $\bar n\approx 2.05$ by $k=3$, which was described as a first step toward an “entangled laser” [1901.00122].

Entanglement and nonclassical correlations can increase under symmetric subtraction. For pure symmetric photon-subtracted TMSV states, the Schmidt coefficients are explicit, and both the negativity
$$
\mathcal N=\sum_{j<\ell} s_j s_\ell
$$
and the logarithmic negativity
$$
E_{\mathcal N}=2\log_2\Bigl(\sum_{j=0}^{\infty}s_j\Bigr)
$$
increase with $k$ at fixed $\lambda$ [1901.00122]. At the level of moment criteria, the Agarwal parameter
$$
I=
\frac{\sqrt{\langle (\hat a_s^\dagger)^2\hat a_s^2\rangle
\,\langle (\hat a_i^\dagger)^2\hat a_i^2\rangle}}
{\langle \hat a_s^\dagger\hat a_s\,
\hat a_i^\dagger\hat a_i\rangle}
-1
$$
is negative for all $k$, and the second-order moment matrix $M^{(2,2)}$ has negative determinant and a negative minimal eigenvalue for all $\lambda$ and $k$, certifying non-classical correlations between the modes [1901.00122].

A general theoretical restriction is that photon subtraction does not produce nonclassical fields from classical fields [2108.12528]. This corrects a common overgeneralization: subtraction is a non-Gaussian filter acting on pre-existing quantum correlations and fluctuations, not a universal generator of nonclassicality from arbitrary classical inputs.

Model dependence remains important. In a one-sided lossy-channel analysis restricted to the subspace $\mathrm{Span}\{\lvert00\rangle,\lvert01\rangle,\lvert10\rangle,\lvert02\rangle,\lvert11\rangle,\lvert20\rangle\}$, one-photon subtraction leaves only populations in $\lvert10\rangle$ and $\lvert20\rangle$, with no off-diagonal coherence in that truncated description and $E_N(\rho_{\rm OPS})=0$ [1906.09592]. This does not contradict the symmetric ideal-state results; it instead shows that entanglement retention depends sensitively on whether subtraction is one-sided or symmetric, on whether loss is present, and on whether one studies the full state or a truncated sector.

## 6. Experimental realizations and neighboring state engineering

The experimentally most mature instance is the simultaneous subtraction from bright two-mode squeezed vacuum, which constitutes the $m_1=m_2=0$ member of the broader PS-TMSFS family. In that implementation, TMSV states were generated in an $8\,\mathrm{mm}$ ppKTP waveguide (type-II SPDC) pumped at $785\,\mathrm{nm}$ with $\sim\!1\,\mathrm{ps}$ pulses from a Ti:Sapphire laser of repetition rate $76\,\mathrm{MHz}$, pulse picked to $229.166\,\mathrm{kHz}$. The signal and idler at $1570\,\mathrm{nm}$ were separated on a PBS, each passed through a $90{:}10$ fiber coupler, and the tapped fractions were monitored by transition-edge sensors (TESs) with approximately $95\%$ detection efficiency; the remaining beams were sent to a second TES pair for conditioned joint-photon-number measurements. The fitted detection parameters were $\eta\approx16.3\%$ and $\nu\approx10^{-3}$ per pulse, and the measured distributions agreed with the lossy theoretical model at the $99\%$-confidence level [1901.00122].

A distinct but closely related experiment used a two-mode squeezed state, linear mixing, and $n$-photon detection to synthesize heralded optical coherent-state superpositions. For two-photon detection with high-efficiency superconducting nanowire single-photon detectors, the freely propagating state exhibited a $67\%$ fidelity with a squeezed even coherent-state superposition of size $\lvert\alpha\rvert^2=3$ [1503.08970]. In the technical formulation underlying that experiment, the heralded non-Gaussian state is the two-component core
$$
\lvert\Psi_n\rangle_{s'}
=
N_n\bigl[\epsilon\sqrt{n(n-1)}\lvert n-2\rangle+\lambda\lvert n\rangle\bigr],
$$
which is then converted by single-mode squeezing into a larger-amplitude squeezed coherent-state superposition [1503.08970].

These experimental results indicate that the practical value of PS-TMSFS-type constructions is not confined to the exact entangled two-mode resource of the most literal definition. The same conditional subtraction architecture supports high-dimensional multiphoton correlation engineering, heralded squeezed-Fock preparation, and cat-state synthesis, with the precise output class determined by the input seed, the entangling transformation, and which modes are retained after conditioning.

## 7. Applications, performance trade-offs, and limitations

A recent phase-space analysis of continuous-variable teleportation with a PS-TMSFS resource examined Braunstein-Kimble teleportation using the characteristic function of the photon-subtracted resource. For coherent-state inputs, the teleportation fidelity takes the form
$$
F_{\rm coh}
=\frac{1}{2\pi}\,\hat F\,a_0\,e^{u^{T}M_4u}\Big|_{u=0},
$$
and for squeezed-vacuum inputs,
$$
F_{\rm sqz}
=\frac{1}{2\pi}\,\hat F\,a_0\,e^{u^{T}M_5u}\Big|_{u=0},
$$
with $M_4$ and $M_5$ specified by the phase-space construction [2606.25471]. The main performance result is restrictive: no substantial enhancement of teleportation fidelity is detected, and fidelity above the classical coherent-state benchmark occurs only for the symmetric $(1,1)$ subtraction configuration in the low-squeezing regime, with that region shrinking as squeezing increases [2606.25471].

This limitation is consistent with the broader resource trade-offs identified in measurement-based squeezed-Fock generation. Photon subtraction increases non-Gaussianity, but it also reduces success probability at higher subtraction orders and can be highly sensitive to optical loss and detector imperfections. In the BS-versus-CZ comparison, uniform photon loss degrades both schemes, with the BS approach remaining more robust to uniform losses, while CZ is more tolerant of detector inefficiency but pays a larger resource cost and requires ancillary squeezing of at least $-15\,\mathrm{dB}$ for $0.99$ fidelity in the reported feasibility estimate [2407.11193].

The resulting picture is nuanced rather than uniform. PS-TMSFS-type states are effective for non-Gaussian state engineering, heralded squeezed-Fock generation, and the manipulation of multiphoton correlations. They do not, however, provide a blanket advantage for all continuous-variable protocols. A plausible implication is that their utility is task-specific: protocols that directly exploit Hermite-structured non-Gaussianity and conditional state preparation can benefit substantially, whereas protocols optimized for Gaussian entanglement, such as standard Braunstein-Kimble teleportation, may not.

Source: https://www.emergentmind.com/topics/photon-subtracted-two-mode-squeezed-fock-state-ps-tmsfs