Open Quantum Materials

An open quantum material is a many-body material or synthetic-matter platform whose relevant states, spectra, or functions depend essentially on coupling to degrees of freedom outside the retained model. Those degrees of freedom may be phonons, photons, substrates, electrical leads, defects, fluctuating fields, measurement channels, or deliberately engineered reservoirs. “Open” is a modeling boundary, not a new chemical category: the same sample can be treated as closed for a femtosecond coherent process and open for transport, relaxation, or steady-state preparation.

We connect environmental coupling to measurable linewidths, conditional dynamics, state preparation and exceptional points. The same complex matrix can describe different experiments, so its physical interpretation matters as much as its eigenvalues. System–Bath Hamiltonians develops the microscopic setup; Driven-Dissipative Matter treats sustained pump–loss phases.

Required background. Driven-Dissipative Matter supplies material pump, loss and steady-response regimes. System–Bath Hamiltonians supplies the retained degrees of freedom, environment and coupling before reduction. Lindblad–GKSL Equation supplies the completely positive Markovian generator and its assumptions. Spectral Functions supplies spectral weights, pole and linewidth interpretation, and their distinction from measured probe intensity.

Materials Coupled to Environments

A useful microscopic starting point is

$$H_{\mathrm{tot}} = H_M + H_E + \sum_\alpha A_\alpha\otimes B_\alpha.$$

$H_M$ contains the material degrees of freedom retained in the calculation, $H_E$ contains the environment, and $A_\alpha\otimes B_\alpha$ specifies what the environment can monitor or exchange. The split must be declared. Treating a cavity photon, optical phonon, or metallic lead as part of $M$ in one model and tracing it into $E$ in another changes both the state space and the observables.

Electrons may exchange energy and momentum with phonons, particles with leads, and radiation with a cavity or the electromagnetic vacuum. Slow spins and fluctuating charges can add memory. Static disorder instead describes uncertainty in a fixed sample; it is not itself a time-dependent reservoir that carries away energy.

Calling all these influences “decoherence” loses important distinctions. A bath can shift levels, broaden poles, thermalize occupations, dephase selected coherences, remove particles, mediate interactions, or create correlated noise. Phonons can destroy electronic phase coherence and also mediate pairing. A cavity can provide loss and also reorganize collective modes. The sign and usefulness of an environmental effect are questions about a specified observable and timescale.

The bath spectrum and state matter as well as the coupling strength. For stationary environmental fluctuations with their mean subtracted, the correlation matrix

$$S_{\alpha\beta}(\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t} \left\langle B_\alpha(t)B_\beta(0) \right\rangle_E$$

determines transition rates and noise in suitable weak-coupling reductions. A Markov approximation requires more than weak coupling: the relevant environmental correlations must decay rapidly compared with the retained evolution, and the system frequencies sampled by the coupling must be resolved consistently. Near a band edge, sharp phonon resonance, cavity pole, or critical environment, memory can be the leading physics.

Choosing a Reduced Description

No single “open Hamiltonian” contains every open-system observable. Different reductions preserve different information.

Observable	Suitable description
Populations and coherences	A density matrix and master equation include mixed states and jump channels.
Quasiparticle poles and lifetimes	A retarded Green function includes spectral weight and self-energy.
Currents through contacts	Scattering or nonequilibrium Green functions include occupations and contacts.
A monitored trajectory	A conditional state includes the selected detector record.
Collective response and noise	A Keldysh theory organizes both response and correlations.
Resonances in a subspace	A projection method retains energy-dependent feedback from eliminated states.

Agreement between two descriptions is conditional. A Lindblad equation can imply a retarded response, but its jump operators contain information that an effective non-Hermitian Hamiltonian alone omits. A retarded self-energy gives linewidths but does not by itself fix occupations; the Keldysh or lesser component carries distribution information.

Self-energy and spectral weight

For a translationally organized material, a retarded single-particle Green function can be written

$$G^R(\mathbf k,\omega) = \left[ \left( \hbar\omega+i0^+ \right)I - H_0(\mathbf k) - \Sigma^R(\mathbf k,\omega) \right]^{-1}.$$

With this energy-denominator convention, the spectral density is

$$\mathcal A(\mathbf k,\omega) = -\frac{1}{\pi} \operatorname{Im} \operatorname{Tr} G^R(\mathbf k,\omega).$$

For a complete set of $N$ canonical fermion orbitals, the full spectral sum is $\int\hbar\,d\omega\,\mathcal A(\mathbf k,\omega)=N$. A plot normalized per unit angular frequency would instead use $\hbar\mathcal A$. Restricting to a low-energy window can omit spectral weight.

The Hermitian part of $\Sigma^R$ shifts and mixes quasiparticle levels. Its anti-Hermitian part produces decay and can mix decay channels. Both may depend strongly on frequency and momentum. The actual pole width follows from the pole equation; it is not generally just minus twice an eigenvalue of $\operatorname{Im}\Sigma^R$ evaluated at one real frequency. Replacing $\Sigma^R(\omega)$ by a constant matrix requires a suitable spectral window.

A complex pole

$$z_n = \varepsilon_n - \frac{i}{2}\Gamma_n$$

has an energy width $\Gamma_n>0$ under the convention $e^{-iz_nt/\hbar}$: amplitude decays as $e^{-\Gamma_n t/(2\hbar)}$ and its squared magnitude as $e^{-\Gamma_n t/\hbar}$. The measured spectral peak need not sit exactly at $\operatorname{Re}z_n$ when backgrounds, overlapping resonances, matrix elements, or frequency-dependent self-energies distort the line shape.

Dissipation as a Design Tool

Environmental coupling becomes a design tool when its spectrum, geometry, or monitored channel is deliberately chosen to stabilize a useful state or response. Examples include cooling selected modes, optically pumping a spin sector, making an entangled state dark, removing defects faster than they are created, or engineering a topological steady-state covariance matrix.

In a Markovian model,

$$\dot\rho = -\frac{i}{\hbar} [H,\rho] + \sum_\mu \mathcal D[L_\mu]\rho.$$

The target $\rho_\star$ must satisfy

$$\mathcal L\rho_\star=0,$$

but a stationary target may coexist with other dark states. Preparation requires attraction from the intended initial states, a usable relaxation time, and realizable couplings. Locality, transport and natural errors can limit preparation even when the target is unique (Diehl and collaborators, 2008; Verstraete, Wolf and Cirac, 2009).

A useful illustration is a two-state rate model. Let $F$ be the target population, $\kappa_p$ the correction rate from the unwanted state, and $\gamma_e$ the rate of leaving the target:

$$\dot F=\kappa_p(1-F)-\gamma_e F, \qquad 1-F_{\mathrm{ss}}=\frac{\gamma_e}{\kappa_p+\gamma_e}.$$

The estimate $1-F_{\mathrm{ss}}\simeq\gamma_e/\kappa_p$ follows only when errors are weak. A general many-body generator cannot be reduced to this formula by simply substituting its smallest decay rate: mode overlaps, nonnormality, dark sectors and correlated errors matter. A global target fidelity can also deteriorate with the number of error locations. Reservoir Engineering develops preparation strategies and their assumptions.

Non-Hermitian Effective Hamiltonians

A non-Hermitian Hamiltonian is usually an effective generator, pole equation, or wave operator. Its physical meaning depends on how it was obtained.

Conditional no-jump evolution

For an initially pure state and ideal counting of every jump channel in a Lindblad model, the unnormalized state conditioned on no detections evolves with

$$H_{\mathrm{eff}} = H - \frac{i\hbar}{2} \sum_\mu L_\mu^\dagger L_\mu.$$

The jump operators include the square roots of their rates. The squared norm obeys

$$\frac{d}{dt} \langle\tilde\psi\vert\tilde\psi\rangle = - \sum_\mu \left\langle L_\mu^\dagger L_\mu \right\rangle_{\tilde\psi}.$$

The remaining squared norm is the no-detection probability; its decrease gives the probability of a jump. Restoring jumps and averaging records recovers trace-preserving evolution. Undetected channels generally leave a mixed conditional state, while homodyne monitoring produces a different, diffusive trajectory. Quantum-Jump Simulation explains how the detection scheme selects the evolution.

Projection and resonance poles

Let $P$ retain a material subspace and $Q=I-P$ contain eliminated states. Outgoing boundary conditions give an energy-dependent operator

$$H_{\mathrm{eff}}(E) ={} PHP + PHQ \frac{1}{ E-QHQ+i0^+ } QHP.$$

The second term is a self-energy. Above an open threshold it acquires an anti-Hermitian part on the real-energy boundary. Resonance poles require analytic continuation of this outgoing resolvent to the appropriate complex-energy sheet and solve

$$\det\!\left[ E-H_{\mathrm{eff}}(E) \right] = 0,$$

so diagonalizing a frozen matrix at one real $E$ can miss energy dependence, extra roots, and threshold structure. Feshbach Projection Formalism develops the state reconstruction and normalization that accompany this reduction.

Phenomenological mode and band models

Classical photonic, acoustic, mechanical, and circuit platforms often produce a non-Hermitian eigenproblem directly after imposing radiation, absorption, gain, or impedance boundary conditions. Electronic quasiparticle models can instead use

$$H_{\mathrm{qp}}(\mathbf k,\omega) = H_0(\mathbf k) + \Sigma^R(\mathbf k,\omega).$$

The microscopic many-body Hamiltonian may remain Hermitian even though the quasiparticle pole operator is not. Complex bands then organize frequency and lifetime, but they are not a list of stationary energies of an isolated crystal.

Left and right eigenvectors

For a diagonalizable non-Hermitian matrix,

$$\begin{aligned} H_{\mathrm{eff}} \lvert R_n\rangle &= z_n\lvert R_n\rangle, \\ \langle L_n\rvert H_{\mathrm{eff}} &= z_n\langle L_n\rvert. \end{aligned}$$

The left and right vectors can be normalized biorthogonally:

$$\langle L_m\vert R_n\rangle = \delta_{mn}.$$

Right eigenvectors alone need not be orthogonal, and the matrix can be highly nonnormal. Small perturbations can then produce large eigenvector and transient-response changes even away from an exact degeneracy.

The biorthogonal quantity $\langle L_n\vert O\vert R_n\rangle$ is mathematically natural, but it is not automatically a laboratory expectation value. Physical observables must be derived from the underlying density matrix, Green function, scattering state, or measurement protocol. This distinction prevents the effective eigenproblem from silently replacing the open-system experiment.

Exceptional Points in Matter

An exceptional point is a parameter value at which eigenvalues and their eigenvectors coalesce, making the effective operator defective. It differs from a Hermitian diabolic point, where degenerate eigenvectors can remain linearly independent and orthogonal.

Two-mode model

Consider two reciprocal modes with complex bare frequencies

$$\Omega_j = \omega_j - \frac{i}{2}\gamma_j$$

and a real conservative coupling $g\geq0$, whose sign is fixed by a mode-phase choice. Here $H_2$ is written in angular-frequency units: $i\partial_t\mathbf a=H_2\mathbf a$, so the corresponding energy matrix is $\hbar H_2$. Reciprocal dissipative coupling can also be complex and is not included in this example:

$$H_2 = \begin{pmatrix} \Omega_1 & g \\ g & \Omega_2 \end{pmatrix}.$$

The eigenvalues are

$$z_\pm = \frac{\Omega_1+\Omega_2}{2} \pm \sqrt{ g^2 + \left( \frac{\Omega_1-\Omega_2}{2} \right)^2 }.$$

The two eigenvalues coincide when the discriminant vanishes,

$$g^2 + \left( \frac{\Omega_1-\Omega_2}{2} \right)^2 = 0.$$

This repeated eigenvalue is an exceptional point only if $H_2$ is not a scalar multiple of the identity. For this model, $g\ne0$ ensures that the zero-discriminant matrix is defective. The case $g=0$ and $\Omega_1=\Omega_2$ instead has two independent eigenvectors.

If $\omega_1=\omega_2$ and the loss rates are unequal, the exceptional point occurs at

$$g = \frac{ \lvert\gamma_1-\gamma_2\rvert }{4}.$$

The nonzero coupling then leaves a one-dimensional eigenspace. Near a generic second-order exceptional point, a perturbation $\epsilon$ produces

$$z_+-z_- \propto \sqrt{\epsilon}.$$

Because the discriminant is complex, two real conditions generally have to be satisfied. A generic second-order exceptional point therefore has codimension two unless a symmetry or constraint reduces the tuning requirement.

Branch topology and encircling

The square root gives two sheets. Encircling the exceptional point once in a two-parameter plane exchanges $z_+$ and $z_-$. A second loop returns the eigenvalue branch; the eigenvector may acquire an additional geometric phase depending on the path and normalization.

Local square-root model $w_\pm=z_\pm-z_{\mathrm{EP}}=\pm\sqrt{\lambda}$. As the discriminant follows $\lambda=re^{i\theta}$ through $0\leq\theta\leq2\pi$, the two eigenvalues exchange their initial positions. The loop does not pass through the degeneracy $\lambda=0$. This is analytic continuation of eigenvalues, not a prediction that a slowly evolving physical state follows the same path.

Static analytic continuation and dynamical encircling must be separated. In a lossy system, arbitrarily slow evolution need not follow an instantaneous eigenstate: differential decay and Stokes transitions can select one mode and produce direction-dependent transfer. Experiments by Doppler and collaborators and by Xu and collaborators demonstrated asymmetric mode conversion in microwave and optomechanical settings. Their dynamical outcomes contain the exceptional-point topology together with nonadiabatic open-system evolution.

Exceptional points from material lifetimes

Interactions and disorder give orbital-dependent self-energies. When two quasiparticle sectors have different lifetimes, the anti-Hermitian part of the self-energy can turn an ordinary band crossing into exceptional points connected by a locus where the real parts of the poles coincide. Kozii and Fu developed this finite-lifetime quasiparticle mechanism; dynamical mean-field calculations by Nagai and collaborators found corresponding bulk-arc structures in heavy-fermion models.

The inference remains spectral and model dependent. An intensity ridge that resembles an arc is not sufficient. One should reconstruct complex poles or a controlled self-energy, verify the defective points at the arc ends, and exclude matrix-element zeros, surface states, finite resolution, and ordinary lifetime-broadened crossings.

Boundaries and the skin effect

Non-Hermitian lattice spectra can depend strongly on boundary conditions. Under the non-Hermitian skin effect, an extensive number of right eigenmodes accumulate near a boundary, so periodic Bloch bands may fail to predict an open sample. A point-gap winding in one dimension can be defined by

$$\nu(E_0) = \frac{1}{2\pi i} \int_{\mathrm{BZ}} dk\, \partial_k \log\det\!\left[ H(k)-E_0 \right].$$

This invariant is meaningful only when the reference point $E_0$ lies outside the periodic complex spectrum. Open-boundary predictions may require a generalized, non-Bloch Brillouin zone (Kunst and collaborators, 2018; Bergholtz, Budich and Kunst, 2021). Boundary localization in a pumped or lossy experiment must also be distinguished from ordinary attenuation, source placement, defects, and finite propagation length.

Experimental Cautions

A split peak is not an exceptional point

An avoided crossing in frequency, a crossing in linewidth, or a change in intensity can suggest a two-mode non-Hermitian model. Stronger evidence tracks both real and imaginary pole components and tests eigenvector behavior.

Scattering poles and zeros are distinct. Coherent perfect absorption concerns a zero of a scattering eigenvalue; lasing concerns a pole reaching the real axis. Either can interact with exceptional structures, but neither is identical to an exceptional point.

Fit the complex response

Intensity-only spectra discard phase and can merge nearby modes. Complex scattering amplitudes, homodyne quadratures or time-domain ringdown provide more information. Fit the instrument response and backgrounds as well as the modes, and compare against a nondegenerate two-pole model over multiple fitting windows. Near a defective point, eigenvectors are ill conditioned and parameter uncertainties can be strongly correlated.

Responsivity is not precision

The square-root splitting $\Delta z\propto\sqrt{\epsilon}$ gives large eigenvalue responsivity to a small perturbation. Metrological precision also depends on linewidth, noise, integration time, estimator bias, technical drift, and the resources needed to hold the device near the exceptional point.

Chen and collaborators (2017) demonstrated enhanced splitting near an exceptional point. In the linear two-mode sensing models analyzed by Lau and Clerk (2018), noise and reciprocity constrain the attainable measurement rate; proximity to an exceptional point alone does not guarantee an advantage. Compare uncertainty in the inferred perturbation at fixed input resources, bandwidth, integration time and detection efficiency (see also Langbein, 2018).

Exercises

1. A Lorentzian pole

For

$$G^R(E) = \frac{1}{ E-\varepsilon+i\Gamma/2 },$$

derive the spectral function $A(E)=-(1/\pi)\operatorname{Im}G^R(E)$ and show that its integral over $E$ is one.

Solution

Multiplying by the complex conjugate denominator gives

$$G^R(E) = \frac{ E-\varepsilon-i\Gamma/2 }{ \left( E-\varepsilon \right)^2 + \left( \Gamma/2 \right)^2 }.$$

Therefore

$$A(E) = \frac{1}{\pi} \frac{ \Gamma/2 }{ \left( E-\varepsilon \right)^2 + \left( \Gamma/2 \right)^2 }.$$

Using $\int_{-\infty}^{\infty}dx\,a/(x^2+a^2)=\pi$ gives $\int dE\,A(E)=1$. A frequency-dependent self-energy redistributes weight between the pole and other spectral contributions. The full canonical spectral sum remains fixed, whereas a restricted energy window need not contain unit weight.

2. No-jump survival

A two-level emitter has $L=\sqrt{\gamma}\lvert g\rangle\langle e\rvert$ and $H=0$. Starting in $\lvert e\rangle$, find the unnormalized no-jump state and its squared norm.

Solution

Since

$$L^\dagger L = \gamma \lvert e\rangle\langle e\rvert,$$

the conditional generator is

$$H_{\mathrm{eff}} = -\frac{i\hbar\gamma}{2} \lvert e\rangle\langle e\rvert.$$

Thus

$$\lvert\tilde\psi(t)\rangle = e^{-\gamma t/2} \lvert e\rangle, \qquad \langle\tilde\psi(t)\vert\tilde\psi(t)\rangle = e^{-\gamma t}.$$

The squared norm is the probability that no photon has been detected up to time $t$ under the ideal monitoring assumed here. The unconditional state also contains the ground-state population created by jump records.

3. Two-mode exceptional point

For equal bare frequencies $\omega_1=\omega_2=\omega_0$, derive the exceptional-point condition with the convention $g\geq0$ and unequal loss rates. Explain why the equal-loss, zero-coupling limit is not exceptional.

Solution

The complex-frequency difference is

$$\Omega_1-\Omega_2 = -\frac{i}{2} \left( \gamma_1-\gamma_2 \right).$$

The discriminant becomes

$$g^2 - \frac{ \left( \gamma_1-\gamma_2 \right)^2 }{16}.$$

It vanishes at

$$g = \frac{ \lvert\gamma_1-\gamma_2\rvert }{4}.$$

Because the losses differ, this coupling is nonzero and $H_2-zI$ has rank one at the repeated eigenvalue $z$. Its eigenspace is therefore one-dimensional: the point is exceptional. If the losses are equal, the zero-discriminant condition instead gives $g=0$ and $H_2=(\omega_0-i\gamma/2)I$, which has a two-dimensional eigenspace. Equality of eigenvalues alone is not sufficient.

4. Branch exchange

Let $z_\pm(\lambda)=\pm\sqrt{\lambda}$ and take $\lambda=re^{i\theta}$. What happens as $\theta$ advances from $0$ to $2\pi$, and then to $4\pi$?

Solution

Choose

$$z_\pm = \pm \sqrt r\, e^{i\theta/2}.$$

After one loop,

$$e^{i(\theta+2\pi)/2} = -e^{i\theta/2},$$

so $z_+$ continues onto $z_-$ and vice versa. After a second loop the eigenvalue returns to its original branch. The associated eigenvector can acquire an additional geometric phase; that phase must be calculated with a declared normalization and path.

5. A slow correction process

In the two-state rate model, suppose $\kappa_p(L)=D/L^2$ and the total error rate $\gamma_e$ is size independent. Find the error-free preparation time and the stationary infidelity with errors. Why should the large-$L$ relaxation time not be confused with successful preparation?

Solution

With $\gamma_e=0$, $1-F(t)=[1-F(0)]e^{-Dt/L^2}$, so a fixed accuracy requires a time proportional to $L^2/D$. With errors,

$$1-F_{\mathrm{ss}}=\frac{\gamma_e L^2}{D+\gamma_e L^2}, \qquad \tau_{\mathrm{relax}}=\frac{1}{D/L^2+\gamma_e}.$$

Weak errors give $1-F_{\mathrm{ss}}\simeq\gamma_e L^2/D$. At large $L$, the relaxation time approaches $1/\gamma_e$, but the stationary fidelity approaches zero: the system reaches a bad state quickly. This two-state example does not establish the preparation scaling of a particular many-body protocol.

For current questions about interacting systems and sensing performance, see Non-Hermitian and PT-symmetric frontiers and Dissipation Engineering.

References

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