Displacement operator

In quantum optics and quantum mechanics, the displacement operator is a unitary operator that shifts the expectation values of the position and momentum quadratures (or equivalently, the annihilation and creation operators) of a quantum harmonic oscillator state in phase space, without altering the state's variance or higher-order moments. Mathematically, it is defined as \hat{D}(\alpha) = \exp(\alpha \hat{a}^\dagger - \alpha^* \hat{a}), where \alpha is a complex parameter encoding the displacement amplitude and phase, and \hat{a}^\dagger, \hat{a} are the bosonic creation and annihilation operators satisfying [\hat{a}, \hat{a}^\dagger] = 1. This operator, formalized by Roy J. Glauber in his seminal work on the quantum theory of optical coherence, generates coherent states—the quantum analogs of classical electromagnetic waves—by acting on the vacuum state: |\alpha\rangle = \hat{D}(\alpha) |0\rangle. Coherent states are right eigenstates of the annihilation operator (\hat{a} |\alpha\rangle = \alpha |\alpha\rangle) and exhibit minimum uncertainty in the quadratures, Poissonian photon-number statistics, and Gaussian Wigner functions centered at the displaced position in phase space. These properties make them ideal for modeling laser light and other coherent radiation fields, bridging classical and quantum descriptions of light. Key properties of the displacement operator include its unitarity (\hat{D}^\dagger(\alpha) = \hat{D}(-\alpha) = \hat{D}^{-1}(\alpha)) and the Baker-Campbell-Hausdorff relation for composition: \hat{D}(\alpha) \hat{D}(\beta) = \exp\left[\frac{1}{2} (\alpha \beta^* - \alpha^* \beta)\right] \hat{D}(\alpha + \beta). It transforms operators via conjugation, such as \hat{D}^\dagger(\alpha) \hat{a} \hat{D}(\alpha) = \hat{a} + \alpha, which displaces the mean field while preserving the vacuum fluctuations. In phase-space formulations, it facilitates representations like the Wigner function. The displacement operator has broad applications in quantum information science, including the engineering of non-classical states for quantum computing and sensing, as well as in optomechanics for cooling mechanical resonators by displacing cavity fields. Experimentally, it has been realized using interferometers to displace coherent states in quantum communication protocols.^[1] Generalizations extend to multi-mode systems, time-dependent drives, and non-harmonic potentials, underscoring its foundational role in modern quantum technologies.

Definition and Formulation

Operator Definition

The quantum harmonic oscillator serves as a foundational model in quantum mechanics, with its Hamiltonian given by

H = \hbar \omega \left( a^\dagger a + \frac{1}{2} \right),

where \hbar is the reduced Planck's constant, \omega is the angular frequency, and a and a^\dagger denote the annihilation and creation operators, respectively, obeying the bosonic commutation relation [a, a^\dagger] = 1. Readers are assumed to be familiar with these operators, which arise from the canonical quantization of the classical harmonic oscillator.^[2] The single-mode displacement operator D(\alpha), introduced in the study of coherent states for the radiation field, is defined as the exponential

D(\alpha) = \exp\left( \alpha a^\dagger - \alpha^* a \right),

where \alpha \in \mathbb{C} is a complex parameter that specifies the magnitude and phase of the displacement. This unitary operator generates translations in the operator algebra of the oscillator.^[3] Acting on the vacuum state |0\rangle, which satisfies a |0\rangle = 0, the displacement operator produces a coherent state |\alpha\rangle = D(\alpha) |0\rangle. This state minimizes the uncertainty in position and momentum quadratures while maintaining the canonical commutation relations, effectively shifting the origin of the phase space.^[3] In the quadrature phase space, defined by the operators X = a + a^\dagger and P = -i(a - a^\dagger) (with [X, P] = 2i), the displacement corresponds to a shift of the expectation values \langle X \rangle = 2 \operatorname{Re}(\alpha) and \langle P \rangle = 2 \operatorname{Im}(\alpha), resulting in a Euclidean displacement distance of $2|\alpha| from the vacuum origin. The vacuum uncertainty ellipse, with variance 1 in both quadratures, is preserved under this translation.

Relation to Coherent States

Coherent states are fundamental quantum states of the harmonic oscillator that arise directly from the action of the displacement operator on the vacuum state. Specifically, the coherent state |\alpha\rangle, parameterized by a complex number \alpha, is defined as |\alpha\rangle = D(\alpha) |0\rangle, where D(\alpha) is the displacement operator and |0\rangle is the vacuum state. This state satisfies the eigenvalue equation for the annihilation operator: a |\alpha\rangle = \alpha |\alpha\rangle, highlighting its role as a right eigenstate of a with complex eigenvalue \alpha. This property underscores the displacement operator's ability to shift the quantum state in phase space while preserving the minimal uncertainty characteristic of Gaussian wave packets. The family of all coherent states \{ |\alpha\rangle \} forms an overcomplete basis for the infinite-dimensional Hilbert space of the harmonic oscillator. This overcompleteness is formalized by the resolution of the identity operator:

\int \frac{d^2 \alpha}{\pi} \, |\alpha\rangle \langle \alpha| = \hat{1},

where the integral is over the complex plane, and d^2 \alpha = d(\operatorname{Re} \alpha) \, d(\operatorname{Im} \alpha). This relation allows any quantum state to be expanded in the coherent state basis, providing a powerful tool for phase-space representations in quantum optics and beyond. The overcomplete nature reflects the redundancy inherent in continuous-variable systems, enabling efficient approximations and calculations. Under the free evolution of the harmonic oscillator Hamiltonian H = \hbar \omega (a^\dagger a + 1/2), coherent states exhibit classical-like behavior. The time-evolved state is given by

|\alpha(t)\rangle = e^{-i \omega t / 2} D(\alpha e^{-i \omega t}) |0\rangle,

which corresponds to a rotation of the displacement parameter \alpha in the complex plane at angular frequency \omega, up to a global phase. This evolution preserves the coherent state's minimal uncertainty and Gaussian profile, mimicking the periodic motion of a classical oscillator. A hallmark of coherent states is their photon number distribution, which follows a Poissonian statistics. The expectation value of the number operator \hat{n} = a^\dagger a is \langle \hat{n} \rangle = |\alpha|^2, and the variance is \Delta \hat{n}^2 = |\alpha|^2, matching the mean and indicating sub-Poissonian noise relative to thermal states but with equal mean and variance characteristic of classical-like fields. This property makes coherent states ideal models for laser light, where intensity fluctuations are minimal.

Mathematical Properties

Unitary Properties

The displacement operator D(\alpha) = \exp(\alpha a^\dagger - \alpha^* a), where a and a^\dagger are the annihilation and creation operators satisfying [a, a^\dagger] = 1, is unitary.^[4] This unitarity follows from the fact that the argument of the exponential is anti-Hermitian, ensuring D^\dagger(\alpha) = D(-\alpha) and D^\dagger(\alpha) D(\alpha) = I.^[4] Consequently, D(\alpha) preserves the norm of quantum states and the inner products between them, making it a valid quantum evolution operator. In the Heisenberg picture, the displacement operator conjugates the bosonic operators to effect a translation:

D^\dagger(\alpha) a D(\alpha) = a + \alpha, \quad D^\dagger(\alpha) a^\dagger D(\alpha) = a^\dagger + \alpha^*.

These relations demonstrate that D(\alpha) shifts the expectation values of the field amplitudes by the complex parameter \alpha, without altering their quantum fluctuations.^[4] The transformed operators retain the canonical commutation relation

[D^\dagger(\alpha) a D(\alpha), D^\dagger(\alpha) a^\dagger D(\alpha)] = 1,

as unitary transformations preserve the algebraic structure of the operator algebra. This conjugation property interprets the displacement operator as a Weyl operator, implementing a translation in phase space.^[4] Specifically, applying D(\alpha) shifts the center of a state's Wigner quasiprobability distribution by amounts proportional to \operatorname{Re}(\alpha) in position and \operatorname{Im}(\alpha) in momentum, while leaving the distribution's shape invariant. Coherent states, obtained by displacing the vacuum |\alpha\rangle = D(\alpha) |0\rangle, exemplify this translation, centering the Gaussian wavepacket at the classical phase-space point corresponding to \alpha.^[4]

Composition and Baker-Hausdorff Formula

The composition of two displacement operators D(\alpha) and D(\beta) is given by the relation

D(\alpha) D(\beta) = \exp\left[\frac{\alpha \beta^* - \alpha^* \beta}{2}\right] D(\alpha + \beta),

where the exponential factor is a phase due to the non-commutativity of the underlying creation and annihilation operators. This formula reveals that the product of displacements corresponds to a total displacement \alpha + \beta, up to a phase factor \exp\left[\frac{\alpha \beta^* - \alpha^* \beta}{2}\right], which arises from the canonical commutation relations [a, a^\dagger] = 1. To derive this, define A = \alpha a^\dagger - \alpha^* a and B = \beta a^\dagger - \beta^* a, so D(\alpha) = e^A and D(\beta) = e^B. The commutator is [A, B] = \alpha \beta^* - \alpha^* \beta, a c-number that commutes with both A and B. The Baker-Campbell-Hausdorff formula then simplifies to e^A e^B = \exp\left(A + B + \frac{1}{2}[A, B]\right), yielding the product rule directly. This multiplication law underscores the non-Abelian structure of the displacement algebra, which generates the Heisenberg-Weyl group—a Lie group whose unitary representations describe the symmetries of the quantum harmonic oscillator in phase space. The phase factor in the composition highlights the group's non-commutativity, distinguishing it from classical translations and enabling applications in quantum state manipulation.

Alternative Representations

Exponential Form Variations

The displacement operator D(\alpha) in quantum optics admits several equivalent exponential representations, distinguished by the ordering of the creation (a^\dagger) and annihilation (a) operators. These variations arise from the non-commutativity of a and a^\dagger, with [a, a^\dagger] = 1, and are derived using the Baker-Campbell-Hausdorff (BCH) formula to disentangle the exponentials. The symmetrically ordered form, D(\alpha) = \exp(\alpha a^\dagger - \alpha^* a), serves as the canonical expression, as it directly shifts the expectation values of the quadrature operators without altering higher-order moments.^[5] Applying the BCH formula to the symmetric form yields the normal-ordered variant, D(\alpha) = \exp\left(-\frac{|\alpha|^2}{2}\right) \exp(\alpha a^\dagger) \exp(-\alpha^* a), where creation operators appear to the left of annihilation operators. This equivalence holds because the commutator [\alpha a^\dagger, -\alpha^* a] = -|\alpha|^2 is a c-number, allowing the exponentials to be separated with a multiplicative prefactor. The normal-ordered form is particularly advantageous for computations involving coherent states |\alpha\rangle = D(\alpha) |0\rangle, as it simplifies the evaluation of normally ordered expectation values, such as those in the Glauber-Sudarshan P-representation, by aligning with the vacuum state's annihilation properties.^[5]^[6] Conversely, the anti-normal-ordered form reverses the operator sequence: D(\alpha) = \exp\left(-\frac{|\alpha|^2}{2}\right) \exp(-\alpha^* a) \exp(\alpha a^\dagger). This representation is obtained analogously via BCH by reordering the terms in the symmetric exponential, placing annihilation operators to the left. It proves useful for anti-normally ordered calculations, including the evaluation of expectation values with the Husimi Q-function, where the operator acts on bras rather than kets, facilitating overlaps with the vacuum in phase-space formulations.^[7]^[6] These exponential variations, while equivalent through BCH relations detailed in the operator's mathematical properties, enable tailored applications in quantum optical simulations and state preparation, prioritizing conceptual clarity over direct numerical expansion.^[5]

Integral and Series Expansions

The displacement operator D(\alpha) admits a power series expansion directly from its exponential definition,

D(\alpha) = \sum_{n=0}^{\infty} \frac{1}{n!} (\alpha a^\dagger - \alpha^* a)^n,

where a and a^\dagger are the annihilation and creation operators, respectively. This series form, while formal, facilitates analytical manipulations in perturbation theory and operator algebra within quantum optics.^[3] More practical for computations, however, are expansions leveraging the orthogonality of the number basis, where explicit matrix elements incorporate associated Laguerre polynomials. The matrix elements of D(\alpha) in the Fock (number) basis \{|n\rangle\} are given by

\langle m | D(\alpha) | n \rangle = \sqrt{\frac{n!}{m!}} \, \alpha^{m-n} \, e^{-|\alpha|^2/2} \, L_n^{m-n}(|\alpha|^2)

for m \geq n, with L_k^l(x) denoting the associated Laguerre polynomial of degree k and order l. For m < n, the expression is obtained by interchanging m and n, replacing \alpha with -\alpha^*, and using L_m^{n-m}(|\alpha|^2). These elements arise from applying the generating function for Laguerre polynomials to the action of D(\alpha) on number states and are essential for calculating overlaps and expectation values in the number basis.^[8] Integral representations of D(\alpha) prove useful for phase-space analyses and deriving generating functions, particularly through Fourier transforms over the complex plane or coherent state overcompleteness relations. These integrals often simplify evaluations of traces or convolutions in quantum optical systems. Displaced number states, defined as |n, \alpha\rangle = D(\alpha) |n\rangle, expand in the number basis using the above matrix elements:

|n, \alpha\rangle = \sum_{m=0}^\infty \langle m | D(\alpha) | n \rangle \, |m\rangle.

For fixed n, this series terminates effectively due to the polynomial nature of the Laguerre functions, providing a finite sum for practical truncation in computations of photon statistics or wavefunctions. This expansion highlights the blending of coherent and number-like features in the state, central to modeling nonclassical light.^[8]

Generalizations

Multimode Displacement Operator

The multimode displacement operator generalizes the single-mode displacement operator to systems comprising multiple independent bosonic modes, such as those encountered in quantum optics for describing multi-mode electromagnetic fields or quantized fields in cavities. For N independent modes, it takes the tensor product form

D(\boldsymbol{\alpha}) = \bigotimes_{k=1}^N D(\alpha_k),

where \boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_N) with each \alpha_k \in \mathbb{C}, and D(\alpha_k) = \exp(\alpha_k a_k^\dagger - \alpha_k^* a_k) denotes the single-mode displacement operator acting on the k-th mode with annihilation operator a_k. Since the mode operators commute across distinct modes, this is equivalent to the exponential of the sum

D(\boldsymbol{\alpha}) = \exp\left( \sum_{k=1}^N (\alpha_k a_k^\dagger - \alpha_k^* a_k) \right).

Acting on the multimode vacuum |\mathbf{0}\rangle = \bigotimes_{k=1}^N |0\rangle_k, the operator generates a separable product of single-mode coherent states

D(\boldsymbol{\alpha}) |\mathbf{0}\rangle = |\boldsymbol{\alpha}\rangle = \bigotimes_{k=1}^N |\alpha_k\rangle.

Displacement operators on different modes commute, satisfying [D(\alpha_j), D(\beta_k)] = 0 for j \neq k, as a direct consequence of the bosonic commutation relations [a_j, a_k^\dagger] = 0 and [a_j, a_k] = 0 for j \neq k; this ensures that the multimode operator preserves the factorized structure of separable states. In configurations involving identical modes, such as symmetric multi-mode setups, collective displacements can be implemented by applying the operator to a single mode while tensoring with identities on the rest, e.g., D(\alpha) \otimes I \otimes \cdots \otimes I, or by coupling modes via beam splitters, which effectively distribute a single-mode displacement into correlated multi-mode coherent states while maintaining overall separability for input vacua.^[9]

Displacement in Other Systems

The concept of the displacement operator, originally formulated for the harmonic oscillator, has been generalized to other quantum systems through the framework of Lie group representations, as introduced by Perelomov in his seminal work on coherent states for arbitrary Lie groups.^[10] This approach defines generalized coherent states as orbits under group actions, enabling displacement-like operators in non-bosonic settings such as spin systems and fermionic modes. In spin systems, the displacement operator generates SU(2) coherent states, which represent classical-like configurations of angular momentum. These states are obtained by applying the unitary operator D(\theta, \phi) = \exp\left[-i \theta (J_x \sin \phi - J_y \cos \phi)\right] to the highest-weight state |j, j\rangle, where \mathbf{J} = (J_x, J_y, J_z) are the spin-j angular momentum operators satisfying the su(2) algebra [J_x, J_y] = i J_z (cyclic). This operator corresponds to a rotation in spin space, analogous to a phase-space shift in the bosonic case, minimizing uncertainty in directions perpendicular to the mean spin vector.^[10] For fermionic systems, such as those involving Majorana fermions or supersymmetric oscillators, a displacement operator can be defined as D(\alpha) = \exp(\alpha c^\dagger - \alpha^* c), where c and c^\dagger are fermionic annihilation and creation operators obeying the anticommutation relation \{c, c^\dagger\} = 1. Due to the Grassmann nature of the parameters and the nilpotency of fermionic operators ((c^\dagger c)^2 = 0), this exponential truncates to a finite polynomial D(\alpha) = 1 + \alpha c^\dagger - \alpha^* c, and it is unitary when using Grassmann variables, generating coherent states analogous to the bosonic version while preserving the fermionic Fock space structure.^[11] In nonlinear oscillators with anharmonic potentials, such as those perturbed from the harmonic case by terms like \lambda x^4, exact displacement operators do not exist, but approximate forms are derived using perturbation theory or operator methods applied to the Heisenberg equations of motion.^[12] These approximations shift the oscillator's equilibrium position while accounting for the nonlinearity, enabling the study of displaced states in systems like molecular vibrations.^[12]

Applications

In Quantum Optics

In quantum optics, the displacement operator plays a central role in describing the quantum states produced by lasers. The output field of an ideal laser approximates a coherent state, which is generated by applying the displacement operator D(\alpha) to the vacuum state, where the complex parameter \alpha is proportional to the classical field amplitude of the laser. This representation captures the Poissonian photon statistics and minimal uncertainty in the electromagnetic field quadratures characteristic of laser light.^[13] Homodyne detection employs the displacement operator to measure the quadrature components of a weak quantum signal. In this technique, the signal mode is mixed with a strong coherent local oscillator on a beam splitter, effectively applying a displacement D(\alpha) (with \alpha determined by the local oscillator amplitude) to shift the signal's phase-space distribution before quadrature projection and photodetection. This method enables precise tomography of quantum states and verification of non-classical features like squeezing.^[14] The displacement operator is essential for generating non-classical states of light beyond pure coherent states. Displaced squeezed states, formed by D(\alpha) S(\zeta) |0\rangle where S(\zeta) is the squeeze operator, produce amplitude-squeezed light with reduced quantum noise in the intensity fluctuations compared to the coherent state limit.^[15] These states are valuable for applications requiring sub-shot-noise precision, such as gravitational wave detection. Experimentally, controlled displacements are realized using beam splitters to combine a signal with a coherent reference field or electro-optic modulators to imprint phase shifts equivalent to displacements. Such implementations have achieved high fidelities exceeding 99% in experiments since the 1990s, enabling reliable manipulation of optical quantum states in laboratory settings.

In Quantum Information

In continuous-variable (CV) quantum computing, the displacement operator serves as a fundamental Gaussian gate, enabling the manipulation of qumodes within cluster-state architectures. These Gaussian operations, including displacements, squeezing, and phase shifts, form a universal gate set when combined with non-Gaussian elements, allowing for measurement-based quantum computation on entangled multimode Gaussian states. In CV cluster states, displacements facilitate the implementation of arbitrary single-mode operations by adjusting the phase-space position of the qumode, which is particularly efficient in linear-optical setups where feedback control can approximate the operator's action.^[16] The displacement operator plays a central role in bosonic quantum error correction codes, notably the Gottesman-Kitaev-Preskill (GKP) codes, which encode discrete logical qubits into the continuous degrees of freedom of an oscillator to protect against displacement errors in the quadrature basis. In GKP codes, logical Pauli operators correspond to large displacements along the position and momentum axes, while stabilizers are defined by periodic displacement operators that detect small shifts, enabling syndrome extraction via homodyne measurements. This approach enables fault-tolerant quantum error correction against displacement noise, with thresholds typically requiring squeezing levels of around 10 dB or higher. Recent experimental realizations have demonstrated GKP stabilization using feedback-driven displacements, achieving logical qubit lifetimes exceeding physical mode decoherence times by factors of 10 or more.^[17]^[18] Multimode displacement operators are essential for generating entangled Gaussian states used in CV quantum teleportation protocols, where they adjust the shared two-mode squeezed vacuum resource to optimize fidelity. In the standard CV teleportation scheme, classical communication of quadrature measurements triggers a corrective displacement on the receiver's mode, enabling faithful transfer of unknown coherent states with fidelities surpassing the classical limit of 0.5 for any non-zero squeezing in the EPR resource, with higher squeezing yielding greater fidelities. For multimode extensions, such as in network-based quantum repeaters, displacements across multiple qumodes enhance entanglement distribution, though fidelity is limited by excess noise to around 0.8-0.9 in current implementations as of 2025. Post-2020 advances in hybrid discrete-continuous variable schemes leverage the displacement operator for quantum transduction between photonic and atomic systems, bridging superconducting qubits with optical modes via coherent state transfers. For instance, displacement-based interfaces in cavity-QED setups enable bidirectional mapping of bosonic excitations, with experimental efficiencies up to around 50% as of 2025 and theoretical proposals exceeding 90% in antiferromagnetic systems while preserving quantum coherence for entanglement swapping.^[19] As of 2025, enhanced realizations in circuit QED have improved GKP code performance for scalable quantum processors.^[20]