Showing papers on "Coherent states published in 2008"
TL;DR: The complete reconstruction and pictorial representation of a variety of radiation states trapped in a cavity in which several photons survive long enough to be repeatedly measured is reported.
Abstract: The state of a microscopic system encodes its complete quantum description, from which the probabilities of all measurement outcomes are inferred. Being a statistical concept, the state cannot be obtained from a single system realization, but can instead be reconstructed from an ensemble of copies through measurements on different realizations. Reconstructing the state of a set of trapped particles shielded from their environment is an important step in the investigation of the quantum-classical boundary. Although trapped-atom state reconstructions have been achieved, it is challenging to perform similar experiments with trapped photons because cavities that can store light for very long times are required. Here we report the complete reconstruction and pictorial representation of a variety of radiation states trapped in a cavity in which several photons survive long enough to be repeatedly measured. Atoms crossing the cavity one by one are used to extract information about the field. We obtain images of coherent states, Fock states with a definite photon number and 'Schrodinger cat' states (superpositions of coherent states with different phases). These states are equivalently represented by their density matrices or Wigner functions. Quasi-classical coherent states have a Gaussian-shaped Wigner function, whereas the Wigner functions of Fock and Schrodinger cat states show oscillations and negativities revealing quantum interferences. Cavity damping induces decoherence that quickly washes out such oscillations. We observe this process and follow the evolution of decoherence by reconstructing snapshots of Schrodinger cat states at successive times. Our reconstruction procedure is a useful tool for further decoherence and quantum feedback studies of fields trapped in one or two cavities.
518 citations
TL;DR: This work uses a superconducting phase qubit, which is a close approximation to a two-level spin system, coupled to a microwave resonator, which acts as a harmonic oscillator, to prepare and analyse pure Fock states with up to six photons.
Abstract: In cavity quantum electrodynamics (QED), light–matter interactions between a single emitter (an atom or an atom-like system with discrete energy levels) and a resonant optical cavity are investigated at a fundamental level. Recent advances in solid-state implementations, which offer great design flexibility, have given this field considerable momentum. An outstanding important question has been which features in such a system show true quantum behaviour and cannot be explained with classical models. Hofheinz et al. study a 'circuit' QED system where a superconducting qubit acts as an atom-like two-energy level system and is embedded in a microwave transmission circuit, acting as the optical cavity. They demonstrate in this system the creation of pure quantum states, known as Fock states, which give specific numbers of energy quanta, in this case photons. Fock states with up to six photons are prepared and analysed. The results are important because cavity QED is expected to play a crucial role in the development of quantum information processing and communication applications. A 'circuit' quantum electrodynamics system where a superconducting qubit acts as an atom-like two-energy level system and is embedded in a microwave transmission circuit (acting as the optical cavity) is studied. In this system, it is demonstrated that the creation of pure quantum states, known as Fock states, which give specific numbers of energy quanta, in this case photons. Fock states with up to six photons are prepared and analysed. Spin systems and harmonic oscillators comprise two archetypes in quantum mechanics1. The spin-1/2 system, with two quantum energy levels, is essentially the most nonlinear system found in nature, whereas the harmonic oscillator represents the most linear, with an infinite number of evenly spaced quantum levels. A significant difference between these systems is that a two-level spin can be prepared in an arbitrary quantum state using classical excitations, whereas classical excitations applied to an oscillator generate a coherent state, nearly indistinguishable from a classical state2. Quantum behaviour in an oscillator is most obvious in Fock states, which are states with specific numbers of energy quanta, but such states are hard to create3,4,5,6,7. Here we demonstrate the controlled generation of multi-photon Fock states in a solid-state system. We use a superconducting phase qubit8, which is a close approximation to a two-level spin system, coupled to a microwave resonator, which acts as a harmonic oscillator, to prepare and analyse pure Fock states with up to six photons. We contrast the Fock states with coherent states generated using classical pulses applied directly to the resonator.
496 citations
TL;DR: In this article, the authors studied the semiclassical properties of the Riemannian spin foam models with Immirzi parameter that are constructed via coherent states and showed that the quantum spin foam amplitudes of an arbitrary triangulation are exponentially suppressed if the face spins do not correspond to a discrete geometry.
Abstract: We study the semiclassical properties of the Riemannian spin foam models with Immirzi parameter that are constructed via coherent states. We show that, in the semiclassical limit, the quantum spin foam amplitudes of an arbitrary triangulation are exponentially suppressed if the face spins do not correspond to a discrete geometry. When they do arise from a geometry, the amplitudes reduce to the exponential of i times the Regge action. Remarkably, the dependence on the Immirzi parameter disappears in this limit.
240 citations
TL;DR: In this article, a new method for extracting an errorless secret key in a continuous-variable quantum key distribution protocol, which is based on Gaussian modulation of coherent states and homodyne detection, is proposed.
Abstract: We propose a new method for extracting an errorless secret key in a continuous-variable quantum key distribution protocol, which is based on Gaussian modulation of coherent states and homodyne detection. The crucial novel feature is an eight-dimensional reconciliation method, based on the algebraic properties of octonions. Since the protocol does not use any postselection, it can be proven secure against arbitrary collective attacks, by using well-established theorems on the optimality of Gaussian attacks. By using this new coding scheme with an appropriate signal to noise ratio, the distance for secure continuous-variable quantum key distribution can be significantly extended.
197 citations
TL;DR: A simple description of the most general collective Gaussian attack in continuous-variable quantum cryptography is provided and the asymptotic secret-key rates which are achievable with coherent states, joint measurements of the quadratures and one-way classical communication are analyzed.
Abstract: We provide a simple description of the most general collective Gaussian attack in continuous-variable quantum cryptography. In the scenario of such general attacks, we analyze the asymptotic secret-key rates which are achievable with coherent states, joint measurements of the quadratures and one-way classical communication.
192 citations
TL;DR: An N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites is investigated and a mean-field approximation is derived, based on a coherent state approximation, in reasonable agreement with the full many-particles evolution.
Abstract: We investigate an N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping transition, are analyzed. The mean-field dynamics is found to be in reasonable agreement with the full many-particle evolution.
190 citations
TL;DR: It is demonstrated that the controlled generation of Fock states with up to 15 photons in a microwave coplanar waveguide resonator coupled to a superconducting phase qubit can be described by a master equation where the lifetime of the n-photon Fock state scales as 1/n, in agreement with theory.
Abstract: We demonstrate the controlled generation of Fock states with up to 15 photons in a microwave coplanar waveguide resonator coupled to a superconducting phase qubit. The subsequent decay of the Fock states, due to dissipation, is then monitored by varying the time delay between preparing the state and performing a number-state analysis. We find that the decay dynamics can be described by a master equation where the lifetime of the n-photon Fock state scales as 1/n, in agreement with theory. We have also generated a coherent state in the microwave resonator, and monitored its decay process. We demonstrate that the coherent state maintains a Poisson distribution as it decays, with an average photon number that decreases with the same characteristic decay time as the one-photon Fock state.
153 citations
TL;DR: This work presents a method for characterizing, with arbitrarily high accuracy, any quantum optical process by studying, via homodyne tomography, its effect on a set of coherent states, that is, classical fields produced by common laser sources.
Abstract: The technologies of quantum information and quantum control are rapidly improving, but full exploitation of their capabilities requires complete characterization and assessment of processes that occur within quantum devices. We present a method for characterizing, with arbitrarily high accuracy, any quantum optical process. Our protocol recovers complete knowledge of the process by studying, via homodyne tomography, its effect on a set of coherent states, that is, classical fields produced by common laser sources. We demonstrate the capability of our protocol by evaluating and experimentally verifying the effect of a test process on squeezed vacuum.
150 citations
TL;DR: In this paper, the authors use SU(2) coherent states to develop a notion of semiclassical states for the quantum geometry which allows to implement them weakly, i.e. on average with minimal uncertainty.
Abstract: General relativity can be written as topological BF theory plus a set of second-class constraints. Classically the constraints provide the geometric interpretation of the B variables and reduce BF to general relativity. In the quantum theory these constraints do not commute and thus cannot be imposed strongly. We use SU(2) coherent states to develop a notion of semiclassical states for the quantum geometry which allows to implement them weakly, i.e. on average with minimal uncertainty. Using the spinfoam formalism, this leads to a background independent regularized path integral for quantum gravity whose variables have a transparent geometric interpretation.
139 citations
TL;DR: Analytically the work distribution of a quantum harmonic oscillator with arbitrary time-dependent angular frequency is calculated and the validity of the quantum Jarzynski equality is verified.
Abstract: We calculate analytically the work distribution of a quantum harmonic oscillator with arbitrary time-dependent angular frequency. We provide detailed expressions for the work probability density for adiabatic and nonadiabatic processes, in the limits of low and high temperature. We further verify the validity of the quantum Jarzynski equality.
139 citations
TL;DR: By using extended bosonic coherent states, a technique to solve the Dicke model exactly is proposed in the numerical sense as discussed by the authors, and the accessible system size is two orders of magnitude higher than that reported in the literature.
Abstract: By using extended bosonic coherent states, a technique to solve the Dicke model exactly is proposed in the numerical sense. The accessible system size is two orders of magnitude higher than that reported in the literature. Finite-size scaling for several observables, such as the ground-state energy, Berry phase, and concurrence, is analyzed. A scaling exponent for ground-state energy is found. An existing discrepancy in the scaling exponent of the concurrence is reconciled.
TL;DR: The exact master equation for two, and its generalization to N, harmonic oscillators interacting with a general environment are expected to be useful for the analysis of quantum coherence, entanglement, fluctuations, and dissipation of mesoscopic objects toward the construction of a theoretical framework for macroscopic quantum phenomena.
Abstract: In this paper we derive an exact master equation for two coupled quantum harmonic oscillators interacting via bilinear coupling with a common environment at arbitrary temperature made up of many harmonic oscillators with a general spectral density function. We first show a simple derivation based on the observation that the two harmonic oscillator model can be effectively mapped into that of a single harmonic oscillator in a general environment plus a free harmonic oscillator. Since the exact one harmonic oscillator master equation is available [B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 (1992)], the exact master equation with all its coefficients for this two harmonic oscillator model can be easily deduced from the known results of the single harmonic oscillator case. In the second part we give an influence functional treatment of this model and provide explicit expressions for the evolutionary operator of the reduced density matrix which are useful for the study of decoherence and disentanglement issues. We show three applications of this master equation: on the decoherence and disentanglement of two harmonic oscillators due to their interaction with a common environment under Markovian approximation, and a derivation of the uncertainty principle at finite temperature for a composite object, modeled by two interacting harmonic oscillators. The exact master equation for two, and its generalization to N, harmonic oscillators interacting with a general environment are expected to be useful for the analysis of quantum coherence, entanglement, fluctuations, and dissipation of mesoscopic objects toward the construction of a theoretical framework for macroscopic quantum phenomena.
TL;DR: In this paper, the convergence of the quantum Bose gas dynamics to the Hartree equation was shown for a class of singular interaction potentials including the Coulomb potential, and it was shown that the mean-field limit is a "semi-classical" limit.
Abstract: In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg-picture dynamics of "observables", thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a "semi-classical" limit.
TL;DR: In this article, the authors investigate quantum repeater protocols based upon atomic qubit-entanglement distribution through optical coherent-state communication, and show that using weaker coherent states, high initial fidelities can still be achieved for larger repeater spacing, at the expense of lower entanglement generation rates.
Abstract: We investigate quantum repeater protocols based upon atomic qubit-entanglement distribution through optical coherent-state communication. Various measurement schemes for an optical mode entangled with two spatially separated atomic qubits are considered in order to nonlocally prepare conditional two-qubit entangled states. In particular, generalized measurements for unambiguous state discrimination enable one to completely eliminate spin-flip errors in the resulting qubit states, as they would occur in a homodyne-based scheme due to the finite overlap of the optical states in phase space. As a result, by using weaker coherent states, high initial fidelities can still be achieved for larger repeater spacing, at the expense of lower entanglement generation rates. In this regime, the coherent-state-based protocols start resembling single-photon-based repeater schemes.
TL;DR: In this article, the authors discuss several methods to produce superpositions of optical coherent states (also known as "cat states") and examine how each would perform in a realistic experiment.
Abstract: We discuss several methods to produce superpositions of optical coherent states (also known as “cat states”). Cat states have remarkable properties that could allow them to be powerful tools for quantum information processing and metrology. A number of proposals for how one can produce cat states have appeared in the literature in recent years. We describe these proposals and present a new simulation and analysis of them incorporating practical issues such as photon loss, detector inefficiency, and limited strength of nonlinear interactions. We also examine how each would perform in a realistic experiment.
TL;DR: In this article, the authors studied the performance of the initial product states of n-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear k-body (kn) Hamiltonian.
Abstract: We study the performance of initial product states of n-body systems in generalized quantum metrology protocols that involve estimating an unknown coupling constant in a nonlinear k-body (kn) Hamiltonian. We obtain the theoretical lower bound on the uncertainty in the estimate of the parameter. For arbitrary initial states, the lower bound scales as 1/nk, and for initial product states, it scales as 1/nk−1/2. We show that the latter scaling can be achieved using simple, separable measurements. We analyze in detail the case of a quadratic Hamiltonian (k=2), implementable with Bose-Einstein condensates. We formulate a simple model, based on the evolution of angular-momentum coherent states, which explains the O(n−3/2) scaling for k=2; the model shows that the entanglement generated by the quadratic Hamiltonian does not play a role in the enhanced sensitivity scaling. We show that phase decoherence does not affect the O(n−3/2) sensitivity scaling for initial product states.
TL;DR: The analytical and numerical results predict that the present QND measurement scheme possesses a high sensitivity, which allows for detecting few photons or even single photons in an ultra-high-Q toroidal system.
Abstract: We theoretically investigate a quantum nondemolition (QND) measurement with optical Kerr effect in an ultra-high-Q microtoroidal system. The analytical and numerical results predict that the present QND measurement scheme possesses a high sensitivity, which allows for detecting few photons or even single photons. Ultra-high-Q toroidal microcavity may provide a novel experimental platform to study quantum physics with nonlinear optics at low light levels.
TL;DR: In this paper, the effects of single-photon annihilation on some paradigmatic light states are investigated. And the invariance of coherent states by this operation is demonstrated by providing the first direct verification of their definition as eigenstates of the photon annihilation operator.
Abstract: The operator annihilating a single quantum of excitation in a bosonic field is one of the cornerstones for the interpretation and prediction of the behavior of the microscopic quantum world. Here we present a systematic experimental study of the effects of single-photon annihilation on some paradigmatic light states. In particular, by demonstrating the invariance of coherent states by this operation, we provide the first direct verification of their definition as eigenstates of the photon annihilation operator.
TL;DR: In this article, the effect of self-phase modulation on the parity gate is studied, introducing generating functions for the Wigner function of a modulated coherent state, and it is shown that for a large class of physical implementations of the phase modulation, the quadrature measurement cannot distinguish between odd and even parity.
Abstract: A possible two-qubit gate for optical quantum computing is the parity gate based on the weak Kerr effect. Two photonic qubits modulate the phase of a coherent state, and a quadrature measurement of the coherent state reveals the parity of the two qubits without destroying the photons. This can be used to create so-called cluster states, a universal resource for quantum computing. Here, the effect of self-phase modulation on the parity gate is studied, introducing generating functions for the Wigner function of a modulated coherent state. For materials with non-electromagnetically-induced-transparency-based Kerr nonlinearities, there is typically a self-phase modulation that is half the magnitude of the cross-phase modulation. Therefore, this effect cannot be ignored. It is shown that for a large class of physical implementations of the phase modulation, the quadrature measurement cannot distinguish between odd and even parity. Consequently, weak nonlinear parity gates must be implemented with physical systems where the self-phase modulation is negligible.
TL;DR: Two coherent-state based methods of quantum propagation, namely, coupled coherent states (CCS) and Gaussian-based multiconfiguration time-dependent Hartree (G-MCTDH), are put on the same formal footing, using a derivation from a variational principle in Lagrangian form.
Abstract: In this article, two coherent-state based methods of quantum propagation, namely, coupled coherent states (CCS) and Gaussian-based multiconfiguration time-dependent Hartree (G-MCTDH), are put on the same formal footing, using a derivation from a variational principle in Lagrangian form. By this approach, oscillations of the classical-like Gaussian parameters and oscillations of the quantum amplitudes are formally treated in an identical fashion. We also suggest a new approach denoted here as coupled coherent states trajectories (CCST), which completes the family of Gaussian-based methods. Using the same formalism for all related techniques allows their systematization and a straightforward comparison of their mathematical structure and cost.
TL;DR: The paper provides a systematic account of simple sampling techniques used in the multidimensional quantum dynamical method of coupled coherent states based on a Gaussian distribution and notices that faster convergence is achieved if "compression" of the basis set decreases as the basis size is increased.
Abstract: The paper provides a systematic account of simple sampling techniques used in the multidimensional quantum dynamical method of coupled coherent states. For the sampling techniques based on a Gaussian distribution, it is noticed that faster convergence is achieved if “compression” of the basis set decreases as the basis size is increased. Good results are obtained for the autocorrelation functions of wave packets propagated in Henon-Heiles potentials with up to 32 degrees of freedom. Further test calculations are performed by employing trains of coherent states sampled on the same classical trajectory with successive time delays.
TL;DR: In this article, the concept of classicality in quantum optics was extended to spin states, where a state can be decomposed as a weighted sum of angular momentum coherent states with positive weights.
Abstract: We extend the concept of classicality in quantum optics to spin states. We call a state ``classical'' if its density matrix can be decomposed as a weighted sum of angular momentum coherent states with positive weights. Classical spin states form a convex set $\mathcal{C}$, which we fully characterize for a spin $1∕2$ and a spin 1. For arbitrary spin, we provide ``nonclassicality witnesses.'' For bipartite systems, $\mathcal{C}$ forms a subset of all separable states. A state of two spins $1∕2$ belongs to $\mathcal{C}$ if and only if it is separable, whereas for a spin $1∕2$ coupled to a spin 1, there are separable states which do not belong to $\mathcal{C}$. We show that in general the question whether a state is in $\mathcal{C}$ can be answered by a linear programming algorithm.
TL;DR: The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator.
Abstract: We compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling $\ensuremath{
u}=k/r$. For $r=2$, these states are the ${Z}_{k}$ Read-Rezayi parafermions, whereas for $rg2$ they represent new FQH states. The $r=k+1$ states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling $2/5,3/7,4/9,\dots{}$. We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the $rg2$ Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.
Journal Article•
TL;DR: In this article, the authors present a method to test quantum behavior of quantum information processing devices, such as quantum memories, teleportation devices, channels, and quantum key distribution protocols, in order to reduce the resources for entanglement verification.
Abstract: We present a method to test quantum behavior of quantum information processing devices, such as quantum memories, teleportation devices, channels, and quantum key distribution protocols. The test of quantum behavior can be phrased as the verification of effective entanglement. Necessary separability criteria are formulated in terms of a matrix of expectation values in conjunction with the partial transposition map. Our method is designed to reduce the resources for entanglement verification. A particular protocol based on coherent states and homodyne detection is used to illustrate the method. A possible test for the quantum nature of memories using two nonorthogonal signal states arises naturally. Furthermore, closer inspection of the measurement process in terms of the Stokes operators reveals a security threat for quantum key distribution involving phase reference beams.
TL;DR: In this paper, the eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equa, i.e.
Abstract: Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equa
TL;DR: In this paper, a Fokker-Planck equation for the Wigner function evolution in a noisy Kerr medium (chi(3) nonlinearity) is presented.
Abstract: A Fokker-Planck equation for the Wigner function evolution in a noisy Kerr medium (chi(3) nonlinearity) is presented. We numerically solved this equation taking a coherent state as an initial condition. The dissipation effects are discussed. We provide examples of quantum interference, sub-Planck phase space structures, and Gaussian vs non-Gaussian dynamical evolution of the state. The results also apply to the description of a nanomechanical resonator with an intrinsic Duffing nonlinearity.
TL;DR: In this paper, the dynamics of Bose-Hubbard systems are described in quantum phase space constructed in terms of generalized SU(M) coherent states, and an explicit evolution equation for the generalized Husimi $(Q)$ and Glauber-Sudarshan $(P)$ distributions are derived.
Abstract: The dynamics of $M$-site, $N$-particle Bose-Hubbard systems is described in quantum phase space constructed in terms of generalized $\text{SU}(M)$ coherent states. These states have a special significance for these systems as they describe fully condensed states. Based on the differential algebra developed by Gilmore, we derive an explicit evolution equation for the (generalized) Husimi $(Q)$ and Glauber-Sudarshan $(P)$ distributions. Most remarkably, these evolution equations turn out to be second-order differential equations where the second-order terms scale as $1/N$ with the particle number. For large $N$ the evolution reduces to a (classical) Liouvillian dynamics. The phase-space approach thus provides a distinguished instrument to explore the mean-field many-particle crossover. In addition, the thermodynamic Bloch equation is analyzed using similar techniques.
TL;DR: In this article, the trigonometric and hyperbolic Poschl-Teller potentials are dealt with from the point of view of classical and quantum mechanics, and there is a natural correspondence between the algebraic structure of these two approaches for both kind of potentials.
TL;DR: In this paper, the authors derived an extended Hubbard model describing the dynamics of the impurities in terms of polarons, i.e. impurities dressed by a coherent state of Bogoliubov phonons.
Abstract: We study the transport of ultracold impurity atoms immersed in a Bose?Einstein condensate (BEC) and trapped in a tight optical lattice. Within the strong-coupling regime, we derive an extended Hubbard model describing the dynamics of the impurities in terms of polarons, i.e. impurities dressed by a coherent state of Bogoliubov phonons. Using a generalized master equation based on this microscopic model, we show that inelastic and dissipative phonon scattering results in (i) a crossover from coherent to incoherent transport of impurities with increasing BEC temperature and (ii) the emergence of a net atomic current across a tilted optical lattice. The dependence of the atomic current on the lattice tilt changes from ohmic conductance to negative differential conductance within an experimentally accessible parameter regime. This transition is accurately described by an Esaki?Tsu-type relation with the effective relaxation time of the impurities as a temperature-dependent parameter.
TL;DR: Various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator and general fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed.
Abstract: Various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator. In this special case two parameters are introduced that are sufficient to completely characterize the force protocol. Explicit results for the characteristic function of work and the corresponding probability distribution are provided and discussed for three different types of initial states of the oscillator: microcanonical, canonical, and coherent states. Depending on the choice of the initial state the probability distributions of the performed work may greatly differ. This result in particular also holds true for identical force protocols. General fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed.