Asymmetric Superpositions of Coherent States

• Asymmetric SCS are nonclassical bosonic states formed by combining coherent state components with unequal weights or arbitrary phase differences, enabling detailed quantum state control.
• They enable advanced state engineering techniques such as heralded orthogonalization, probabilistic photon replacement, and projective measurements for seamless continuous- and discrete-variable interfacing.
• Experimental protocols using beam splitters, homodyne detection, and displacement operations achieve high-fidelity state generation while addressing challenges like nonorthogonality and channel noise.

Asymmetric superpositions of coherent states (SCS) are nonclassical bosonic states in which coherent components are combined with unequal weights, nontrivial relative phases, or more general phase-space asymmetries. In the standard two-component setting, the literature uses forms such as

$$|\Phi_{\mathrm{SCS}}\rangle = |\alpha\rangle + c|-\alpha\rangle$$

and

$$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$

with asymmetry carried by the complex coefficient $c$ or by the pair $a,b$. A closely related measurement-oriented formulation expands arbitrary states in the even/odd SCS basis,

$$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$

so that asymmetry may appear as unequal $c_0,c_1$ or as arbitrary phase $\phi$ (Kruse et al., 2017, Molnar et al., 2017, Izumi et al., 2017). Because coherent states remain nonorthogonal,

$$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$

asymmetric SCS lie at the intersection of state engineering, probabilistic discrimination, continuous-variable/discrete-variable interfacing, and noise-adapted bosonic encoding (Kruse et al., 2017).

1. Canonical structure, parity, and what “asymmetric” denotes

In the two-component case, even and odd cat states are the symmetric special cases $|\alpha\rangle \pm |-\alpha\rangle$. The arbitrary-basis construction

$$|\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,\qquad |\pi_1^{\mathrm{SCS}}\rangle = c_1 e^{-i\phi}|C_+\rangle - c_0|C_-\rangle,$$

makes explicit that asymmetry is not limited to unequal magnitudes of the two coherent components: it also includes arbitrary phase and arbitrary bias between the even and odd sectors. In this formulation, “symmetric SCS” corresponds to $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$0, whereas “asymmetric SCS” corresponds to $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$1 or arbitrary $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$2 (Izumi et al., 2017).

Photon-number parity is the defining algebraic distinction between the symmetric limits. For the normalized cat state

$$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$3

the photon-number probability is

$$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$4

Thus, $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$5 yields an even cat with only even photon numbers, and $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$6 yields an odd cat with only odd photon numbers. The same analysis indicates that for a general state $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$7, both even and odd photon-number sectors are generally populated, with weights controlled by the relative amplitudes and phases (Hernández-Sánchez et al., 2023).

A recurrent ambiguity in the literature is that “asymmetry” can refer to different but adjacent notions. Across the cited works, it may denote unequal coefficients in $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$8, arbitrary phase in the $$|\psi_{\mathrm{asymSCS}}\rangle = a|\beta\rangle + b|-\beta\rangle,$$9 basis, multi-component phase-weighted superpositions, or controlled phase-space displacement and squeezing of the constituent states. This broader usage is visible in state-generation, measurement, and higher-dimensional encoding papers (Molnar et al., 2017, Izumi et al., 2017, Podoshvedov, 2011).

2. Heralded orthogonalization and conversion to discrete-variable superpositions

A central obstacle for coherent-state encodings is nonorthogonality. An experimentally feasible route around this is probabilistic orthogonalization by quantum catalysis, or photon replacement. In the protocol, an unknown input $c$0 or $c$1 enters one port of a beam splitter, an ancilla single photon $c$2 enters the other, the transmissivity $c$3 is chosen as a function of $c$4, and a projective photon-number measurement is performed on one output mode. The value of $c$5 is fixed so that the heralded outputs $c$6 and $c$7 become orthogonal; for $c$8, the paper gives the example $c$9 (Kruse et al., 2017).

The operation is heralded and non-destructive in the specific sense that success is announced by the photon-number detection event, while the conditional state in the other mode is left available for further processing. The output states are close approximations of the discrete-variable superpositions

$a,b$0

so the orthogonalized pair provides a direct continuous-variable to discrete-variable interface (Kruse et al., 2017).

For superpositions of coherent states, linearity makes the map especially useful. A general input

$a,b$1

is transformed to

$a,b$2

The symmetric limits $a,b$3 are singled out in the analysis: the even SCS output approximates a weakly squeezed vacuum state with $a,b$4 fidelity, while the odd SCS output approximates a weakly squeezed single-photon Fock state. The paper further notes that the success probability can approach the Ivanovic-Dieks-Peres bound for small $a,b$5, and that cascaded stages with adaptive beam-splitter and measurement settings can be used to increase overall success probability (Kruse et al., 2017).

This establishes a precise sense in which asymmetric SCS can be coherently converted into qubit-like states without direct destructive readout of the encoded superposition. A plausible implication is that such orthogonalization is particularly relevant when asymmetric SCS are used as intermediate bosonic resources rather than terminal measurement states.

3. State engineering and projective measurement

Several works treat asymmetric SCS as directly engineerable targets in traveling optical fields. One approach uses only linear optical elements and homodyne measurements. In one scheme, three 50:50 beam splitters and three homodyne detectors produce superpositions of coherent states on a line or a lattice in phase space; in a second scheme, two beam splitters and two homodyne detectors combine a coherent-state superposition with a squeezed vacuum approximated by equidistant coherent states. In both cases, measurement-induced nonlinearity is supplied by post-selected homodyne outcomes, and the coefficients of the output coherent-state expansion are controlled by input amplitudes, phase shifts, and quadrature outcomes. The paper states that this allows the generation of SCS with arbitrary weights, including

$a,b$6

and more general sums $a,b$7 (Molnar et al., 2017).

The optimization criterion in that framework is the misfit

$a,b$8

The reported minimal misfit values are between $a,b$9 and $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$0, and the total success probability is typically in the $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$1 range for precise state generation. The same paper emphasizes the usual trade-off: broader post-selection windows increase probability but reduce fidelity (Molnar et al., 2017).

A complementary line of work implements projective measurement onto arbitrary SCS bases rather than state preparation. The realized device combines a displacement operation with photon counting,

$$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$2

and uses quantum detector tomography reconstructed from coherent probes. For small $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$3, the target SCS basis is well approximated in the $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$4 sector, so tuning $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$5 rotates the effective measurement basis anywhere on the SCS-basis Bloch sphere, including strongly asymmetric cases. The reported experimental fidelities exceed those of ideal homodyne detection and ideal photon-number-resolving detection over a wide range of $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$6 and $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$7 (Izumi et al., 2017).

A third preparation route alternates photon additions and displacements beginning from a seed coherent state. In that representation, arbitrary one-mode states are expanded in displaced number states $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$8, and suitable sequences of $$|C_\pm\rangle = \frac{|\alpha\rangle \pm |-\alpha\rangle}{\mathcal N_\pm},\qquad |\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,$$9 and $c_0,c_1$0 generate even and odd displaced squeezed SCSs with high fidelity. The paper reports fidelities in the range $c_0,c_1$1 for squeezed or shifted SCS amplitudes up to approximately $c_0,c_1$2, and uses the same machinery to construct local rotations and a Hadamard gate for coherent states. Because the seed amplitude and intermediate displacements determine the final phase-space offset, this scheme naturally produces shifted, and therefore asymmetric, SCS pairs (Podoshvedov, 2011).

More geometric asymmetric constructions place a finite number of coherent states on an ellipse or a lattice and optimize the coefficients and geometry numerically via a genetic algorithm. The misfit is again

$c_0,c_1$3

For target states with elliptical or irregular Wigner structure, ellipse and lattice arrangements outperform more symmetric line or circle placements; for example, the paper gives $c_0,c_1$4 on an ellipse versus $c_0,c_1$5 on a line for a squeezed number state with $c_0,c_1$6 (Adam et al., 2014). This does not define asymmetric SCS in the narrow two-component sense, but it does establish asymmetric coherent-state placement as a systematic state-engineering principle.

4. Multi-component, ququat, qudit, and $c_0,c_1$7-state generalizations

The most explicit higher-dimensional generalization of asymmetric SCS is the ququat encoding

$c_0,c_1$8

where the $c_0,c_1$9 are arbitrary complex coefficients. The associated orthonormal basis is given by the multi-photonic states

$\phi$0

with the defining photon-number property that $\phi$1 has nonzero support only on photon numbers $\phi$2. In this basis, arbitrary four-component asymmetric SCS encode general ququats. The paper further constructs bipartite four-component entangled coherent states by sending these states through a 50:50 beam splitter, and discusses almost perfect teleportation, higher-dimensional BB84, and superdense coding within this encoding (Mishra et al., 2012).

A more general $\phi$3-component family is

$\phi$4

These states have the pseudo-number property that their Fock support is restricted to photon numbers congruent to $\phi$5 modulo $\phi$6. The index $\phi$7 therefore functions as a controlled asymmetry label. Under amplification, this family behaves nontrivially: $\phi$8 tends to show better quantum Fisher information for relatively small amplitudes, $\phi$9 is better in the larger-amplitude regime, and the performance of the two schemes becomes indistinguishable as the amplitude grows sufficiently large. The same analysis concludes that there is no universal best amplification strategy for SCS, because fidelity and gain depend on coherent-state amplitudes, number of components, and relative phases (Jeon et al., 2024).

Finite-dimensional truncations produce an additional qudit perspective. Qudit coherent states may be defined either by a truncated displacement operator acting on vacuum or by truncating the Poissonian expansion of a Glauber coherent state at photon number $$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$0. In that setting, displacing the vacuum in the qudit Hilbert space can generate macroscopically distinguishable superpositions of two qudit coherent states with high fidelity, while the parity of the resulting cat state depends on whether $$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$1 is even or odd. The same work analyzes their Wigner functions, nonclassical volume, and optical tomograms, and relates their preparation to nonlinear and linear quantum scissors (Miranowicz et al., 2013).

Outside quantum optics in the narrow sense, $$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$2-state Schrödinger cat states also appear as

$$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$3

These phase-weighted superpositions transform in irreducible representations of $$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$4, so their asymmetry is encoded in the phase label $$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$5. In the large-$$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$6 limit they approach one-row Young tableau states, while their quantum Fisher information is $$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$7, quantifying localization in the angular direction of phase space (Lin et al., 2020).

5. Discrimination, amplification, and physical signatures

One discrimination strategy does not measure the field directly at all. In a cavity-QED setting with an AC Stark term,

$$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$8

the time-averaged atomic inversion lineshape depends on the photon-number distribution of the field. For low mean photon number, $$|\langle \alpha|-\alpha\rangle| = e^{-2|\alpha|^2}>0,$$9, and significant $|\alpha\rangle \pm |-\alpha\rangle$0, this permits discrimination between even and odd cat states because the two parity-restricted photon-number distributions imprint distinct sets of shifted resonances on the atomic response. The same analysis suggests that for a general asymmetric superposition $|\alpha\rangle \pm |-\alpha\rangle$1, the lineshape reflects a weighted parity mixture; distinguishability is then less sharp and is strongest when the state remains close to a pure even or odd cat (Hernández-Sánchez et al., 2023).

Amplification protocols likewise depend sensitively on parity and asymmetry. Mixing two odd SCSs at a 50:50 beam splitter and post-selecting by homodyne detection can produce a larger even SCS, but high fidelity is obtained only for large input amplitudes. By contrast, using opposite parities—an even SCS and an odd SCS—yields a larger odd SCS with nearly perfect fidelity even for small input amplitude when post-selection is performed near $|\alpha\rangle \pm |-\alpha\rangle$2. The same paper explicitly notes that if the input amplitudes or parities are mismatched, the output SCS can become asymmetric (Oh et al., 2018). This identifies parity engineering as a direct route to producing asymmetric cats rather than merely a method for amplifying symmetric ones.

Asymmetric or multi-component SCS also acquire experimentally visible signatures outside quantum-optical state tomography. In the theory of microwave-induced resistance oscillations, standard coherent states explain the conventional oscillatory magnetoresistance pattern, while even/odd two-component cat states populate only every other Landau level and triangular three-component cats populate every third Landau level. The resulting resonance conditions shift from $|\alpha\rangle \pm |-\alpha\rangle$3 to $|\alpha\rangle \pm |-\alpha\rangle$4 or $|\alpha\rangle \pm |-\alpha\rangle$5, and the same level-selective structure is used to explain magnetoresistance collapse in ultra-high-mobility samples (Inarrea, 2023). The paper formulates the general SCS as a sum $|\alpha\rangle \pm |-\alpha\rangle$6 but does not work out unequal-weight asymmetric cases explicitly. This suggests that, in that transport framework, asymmetry would be encoded through modified Landau-level selection rules rather than through parity alone.

6. Noise adaptation, phase-space geometry, entanglement, and nonstandard settings

A recent protection strategy treats asymmetry as a property of the channel rather than of the coefficients of the cat state itself. In an asymmetric thermal lossy channel, the signal couples through an unbalanced beam splitter to an axis-aligned squeezed thermal environment, and the field Wigner function evolves by Gaussian convolution. The key result is that optimal pre-squeezing of the input SCS compensates the channel asymmetry: the survival of central Wigner negativity is governed by the channel-only condition

$|\alpha\rangle \pm |-\alpha\rangle$7

and the environmental asymmetry $|\alpha\rangle \pm |-\alpha\rangle$8 can be offset by a corresponding squeezing of the input (Provazník et al., 2024). In this usage, “asymmetric SCS” refers to cat states propagating through asymmetric channels rather than to unequal coefficients in the superposition itself.

A broader geometric extension replaces coherent states by arbitrary Gaussian states. For

$|\alpha\rangle \pm |-\alpha\rangle$9

the Wigner function decomposes into two Gaussian peaks plus an interference term. When the constituent Gaussian profiles differ, the interference phase acquires a quadratic form; in one degree of freedom, the phase is hyperbolic rather than linear, so the familiar straight interference fringes of standard symmetric cats are replaced by hyperbolic ones. The paper further shows that this phase-space structure survives the action of a thermal reservoir (Nicacio et al., 2010). A plausible implication is that asymmetry in squeezing or orientation is as operationally significant as asymmetry in coefficient magnitude.

Photon addition creates another asymmetry channel. For entangled photon-added coherent states,

$$|\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,\qquad |\pi_1^{\mathrm{SCS}}\rangle = c_1 e^{-i\phi}|C_+\rangle - c_0|C_-\rangle,$$0

asymmetry appears simultaneously in $$|\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,\qquad |\pi_1^{\mathrm{SCS}}\rangle = c_1 e^{-i\phi}|C_+\rangle - c_0|C_-\rangle,$$1, in $$|\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,\qquad |\pi_1^{\mathrm{SCS}}\rangle = c_1 e^{-i\phi}|C_+\rangle - c_0|C_-\rangle,$$2, and in the photon-addition orders $$|\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,\qquad |\pi_1^{\mathrm{SCS}}\rangle = c_1 e^{-i\phi}|C_+\rangle - c_0|C_-\rangle,$$3. The analysis shows that photon addition acts as an entanglement enhancer for superpositions of coherent states and also increases robustness against depolarization. For maximally entangled states, the critical depolarization value is $$|\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,\qquad |\pi_1^{\mathrm{SCS}}\rangle = c_1 e^{-i\phi}|C_+\rangle - c_0|C_-\rangle,$$4 (Dominguez-Serna et al., 2016).

Two conceptually broader settings further enlarge the meaning of asymmetric SCS. On a Möbius strip, coherent states and their superpositions inherit $$|\pi_0^{\mathrm{SCS}}\rangle = c_0|C_+\rangle + c_1 e^{i\phi}|C_-\rangle,\qquad |\pi_1^{\mathrm{SCS}}\rangle = c_1 e^{-i\phi}|C_+\rangle - c_0|C_-\rangle,$$5 periodicity; nonzero relative phase produces asymmetric SCS while preserving the Möbius-strip periodicity, and minimal-uncertainty SCS are interpreted as cat-state analogues in that topology (Prudêncio et al., 2013). In finite-dimensional optical truncation, superpositions of qudit coherent states generated by displacement of vacuum display phase-space interference, parity selection, nonclassical volume, and directly measurable optical tomograms (Miranowicz et al., 2013).

Taken together, these results show that asymmetric SCS are not a single narrowly defined family but a hierarchy of bosonic superpositions whose asymmetry may reside in coefficients, phases, component number, phase-space displacement, squeezing, photon-addition structure, or channel adaptation. The recurring technical themes are nonorthogonality, parity control, measurement-induced nonlinearity, and the use of phase-space structure as both a design principle and a diagnostic.