Pulsed quantum optomechanics

Studying mechanical resonators via radiation pressure offers a rich avenue for the exploration of quantum mechanical behavior in a macroscopic regime. However, quantum state preparation and especially quantum state reconstruction of mechanical oscillators remains a significant challenge. Here we propose a scheme to realize quantum state tomography, squeezing, and state purification of a mechanical resonator using short optical pulses. The scheme presented allows observation of mechanical quantum features despite preparation from a thermal state and is shown to be experimentally feasible using optical microcavities. Our framework thus provides a promising means to explore the quantum nature of massive mechanical oscillators and can be applied to other systems such as trapped ions.     

Entanglement-assisted concatenated quantum codes

Entanglement-assisted concatenated quantum codes (EACQCs), constructed by concatenating two quantum codes, are proposed. These EACQCs show significant advantages over standard concatenated quantum codes (CQCs). First, we prove that, unlike standard CQCs, EACQCs can beat the nondegenerate Hamming bound for entanglement-assisted quantum error-correction codes (EAQECCs). Second, we construct families of EACQCs with parameters better than the best-known standard quantum error-correction codes (QECCs) and EAQECCs. Moreover, these EACQCs require very few Einstein–Podolsky–Rosen (EPR) pairs to begin with. Finally, it is shown that EACQCs make entanglement-assisted quantum communication possible, even if the ebits are noisy. Furthermore, EACQCs can outperform CQCs in entanglement fidelity over depolarizing channels if the ebits are less noisy than the qubits. We show that the error-probability threshold of EACQCs is larger than that of CQCs when the error rate of ebits is sufficiently lower than that of qubits. Specifically, we derive a high threshold of 47% when the error probability of the preshared entanglement is 1% to that of qubits.

Quantum error-correction codes (QECCs) are necessary to realize quantum communications and to make fault-tolerant quantum computers (1, 2). The stabilizer formalism provides a useful way to construct QECCs from classical codes, but certain orthogonality constraints are required (3). The entanglement-assisted (EA) QECC (EAQECC) (4–6) generalizes the stabilizer code. By presharing some entangled states between the sender (Alice) and the receiver (Bob), EAQECCs can be constructed from any classical linear codes without the orthogonality constraints. Therefore, the construction could be greatly simplified. As an important physical resource, entanglement can boost the classical information capacity of quantum channels (7–12). Recently, it has been shown that EAQECCs can violate the nondegenerate quantum Hamming bound (13) or the quantum Singleton bound (14).Compared to standard QECCs, EAQECCs must establish some amount of entanglement before transmission. This preshared entanglement is the price to be paid for enhanced communication capability. In a sense, we need to consider the net transmission of EAQECCs—i.e., the number of qubits transmitted minus that of ebits preshared. Further, it is difficult to preserve too many noiseless ebits in EAQECCs at present. Thus, we have to use as few ebits as possible to conduct the communication—e.g., one or two ebits are preferable (15–18). In addition, EAQECCs with positive net transmission and little entanglement can lead to catalytic quantum codes (4, 6), which are applicable to fault-tolerant quantum computation (FTQC). In ref. 4, a table of best-known EAQECCs of length up to 10 was established through computer search or algebraic methods. Several EAQECCs in ref. 4 have larger minimum distances than the best-known standard QECCs of the same length and net transmission. However, for larger code lengths, the efficient construction of EAQECCs with better parameters than standard QECCs is still unknown.In classical coding theory, concatenated codes (CCs), originally proposed by Forney in the 1960s (19), provide a useful way of constructing long codes from short ones. CCs can achieve very large coding gains with reasonable encoding and decoding complexity (20). Moreover, CCs can have large minimum distances since the distances of the component codes are multiplied. As a result, CCs have been widely used in many digital communication systems—e.g., the NASA standard for the Voyager program (21) and the compact disc (20). Similarly, in QECCs, the concatenated quantum codes (CQCs), introduced by Knill and Laflamme in 1996 (22), are also effective for constructing good quantum codes. In particular, it has been shown that CQCs are of great importance in realizing FTQC (23–25).Moreover, there exists a specific phenomenon in QECCs, called “error degeneracy,” which distinguishes quantum codes from classical ones in essence. It is widely believed that degenerate codes can correct more quantum errors than nondegenerate ones. Indeed, there are some open problems concerning whether degenerate codes can violate the nondegenerate quantum Hamming bound (26) or can improve the quantum-channel capacity (27, 28). Many CQCs have been shown to be degenerate, even if the component codes are nondegenerate—e.g., Shor’s [[9,1,3]] code and the [[25,1,9]] CQC (23, 29). If we introduce extra entanglement to CQCs, it is possible to improve the error-degeneracy performance of CQCs.In this article, we generalize the idea of concatenation to EAQECCs and propose EACQCs. We show that EACQCs can beat the nondegenerate quantum Hamming bound, while standard CQCs cannot. Several families of degenerate EACQCs that can surpass the nondegenerate Hamming bound for EAQECCs are constructed. The same conclusion could be reached for asymmetric error models, in which the phase-flip errors (Z errors) happen more frequently than the bit-flip errors (X errors) (30, 31). Furthermore, we derive a number of EACQCs with better parameters than the best-known QECCs and EAQECCs. In particular, we see that many EACQCs have positive net transmission, and each of them consumes only one or two ebits. Thus, they give rise to catalytic EACQCs with little entanglement and better parameters than the best-known QECCs. Further, we show that the EACQC scheme makes EA quantum communication possible, even if the ebits are noisy. We compute the entanglement fidelity (EF) of the [[15,1,9;10]] EACQC by using Bowen’s [[3,1,3;2]] EAQECC (32) or the [[3,1,3;2]] EA repetition code (4, 6) as the inner code. The outer code is the standard [[5,1,3]] stabilizer code. We show that the [[15,1,9;10]] EACQC performs much better than the [[25,1,9]] CQC over depolarizing channels if the ebits suffer a lower error rate than the qubits. Moreover, we compute the error-probability threshold of EACQCs, and we show that EACQCs have much higher thresholds than CQCs when the error rate of ebits is sufficiently lower than that of qubits.     

Two-boson quantum interference in time

The celebrated Hong–Ou–Mandel effect is the paradigm of two-particle quantum interference. It has its roots in the symmetry of identical quantum particles, as dictated by the Pauli principle. Two identical bosons impinging on a beam splitter (of transmittance 1/2) cannot be detected in coincidence at both output ports, as confirmed in numerous experiments with light or even matter. Here, we establish that partial time reversal transforms the beam splitter linear coupling into amplification. We infer from this duality the existence of an unsuspected two-boson interferometric effect in a quantum amplifier (of gain 2) and identify the underlying mechanism as time-like indistinguishability. This fundamental mechanism is generic to any bosonic Bogoliubov transformation, so we anticipate wide implications in quantum physics.

The laws of quantum physics govern the behavior of identical particles via the symmetry of the wave function, as dictated by the Pauli principle (1). In particular, it has been known since Bose and Einstein (2) that the symmetry of the bosonic wave function favors the so-called bunching of identical bosons. A striking demonstration of bosonic statistics for a pair of identical bosons was achieved in 1987 in a seminal experiment by Hong, Ou, and Mandel (HOM) (3), who observed the cancellation of coincident detections behind a 50:50 beam splitter (BS) when two indistinguishable photons impinge on its two input ports (Fig. 1A). This HOM effect follows from the destructive two-photon interference between the probability amplitudes for double transmission and double reflection at the BS (Fig. 1B). Together with the Hanbury Brown and Twiss effect (4, 5) and the violation of Bell inequalities (6, 7), it is often viewed as the most prominent genuinely quantum feature: it highlights the singularity of two-particle quantum interference as it cannot be understood in terms of classical wave interference (8, 9). It has been verified in numerous experiments over the last 30 y (see, e.g., refs. 10–13), even in case the single photons are simultaneously emitted by two independent sources (14–16) or within a silicon photonic chip (17, 18). Remarkably, it has even been experimentally observed with 4He metastable atoms, demonstrating that this two-boson mechanism encompasses both light and matter (19).Open in a separate windowFig. 1.(A) If two indistinguishable photons (represented in red and green for the sake of argument) simultaneously enter the two input ports of a 50:50 BS, they always exit the same output port together (no coincident detection can be observed). (B) The probability amplitudes for double transmission (Left) and double reflection (Right) precisely cancel each other when the transmittance is equal to 1/2. This is a genuinely quantum effect, which cannot be described as a classical wave interference. (C) The correlation function exhibits an HOM dip when the time difference Δt between the two detected photons is close to zero (i.e., when they tend to be indistinguishable).Here, we explore how two-boson quantum interference transforms under reversal of the arrow of time in one of the two bosonic modes (Fig. 2A). This operation, which we dub partial time reversal (PTR), is unphysical but nevertheless central as it allows us to exhibit a duality between the linear optical coupling effected by a BS and the nonlinear optical (Bogoliubov) transformation effected by a parametric amplifier. As a striking implication of these considerations, we predict a two-photon interferometric effect in a parametric amplifier of gain 2 (which is dual to a BS of transmittance 1/2). We argue that this unsuspected effect originates from the indistinguishability between photons from the past and future, which we coin “time-like” indistinguishability as it is the partial time-reversed version of the usual “space-like” indistinguishability that is at work in the HOM effect.Open in a separate windowFig. 2.(A) BS under PTR, flipping the arrow of time in mode b^. The PTR duality is illustrated when n photons impinge on port b^ (with vacuum on port â), and we condition on all photons being reflected. The retrodicted state of mode b^′ (initially the vacuum state 0) back propagates from the detector to the source (suggested by a wavy arrow). This yields the same transition probability amplitude (up to a constant) as for a PDC of gain g=1/η with input state 0,0 and output state n,n. PDC is an active Bogoliubov transformation, requiring a pump beam (represented in blue). Note that the PTR duality is rigorously valid when this pump beam is of high intensity (i.e., treated as a classical light beam) since the Hamiltonian HPDC of Eq. 4 holds in this limit only. (B) Operational view of the PTR duality. As noted in ref. 20, if we prepare the entangled (EPR) state Ψb,b′′∝∑n=0∞n,n and send mode b^ in the BS, we get the output state Ψa′,b′′∝∑n=0∞⁡sinn θ n,n, which is precisely the two-mode squeezed vacuum state produced by PDC when the signal and idler modes are initially in the vacuum state.Since Bogoliubov transformations are ubiquitous in quantum physics, it is expected that this two-boson interference effect in time could serve as a test bed for a wide range of bosonic transformations. Furthermore, from a deeper viewpoint, it would be fascinating to demonstrate the consequence of time-like indistinguishability in a photonic or atomic platform as it would help in elucidating some heretofore overlooked fundamental property of identical quantum particles.     

Random matrices and quantum chaos

The theory of random matrices has far-reaching applications in many different areas of mathematics and physics. In this note, we briefly describe the state of the theory and two of the perhaps most surprising appearances of random matrices, namely in the theory of quantum chaos and in the theory of prime numbers.     

The quantum MacMahon Master Theorem

We state and prove a quantum generalization of MacMahon's celebrated Master Theorem and relate it to a quantum generalization of the boson-fermion correspondence of physics.     

Dissipative properties of quantum systems

We consider the dissipative properties of large quantum systems from the point of view of kinetic theory. The existence of a nontrivial collision operator imposes restrictions on the possible collisional invariants of the system. We consider a model in which a discrete level is coupled to a set of quantum states and which, in the limit of a large "volume," becomes the Friedrichs model. Because of its simplicity this model allows a direct calculation of the collision operator as well as of related operators and the constants of the motion. For a degenerate spectrum the calculations become more involved but the conclusions remain simple. The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator. For a class of observables we obtain ergodic behavior, and this reformulation of the ergodic problem may be used in statistical mechanics to study the ergodicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration.     

Imaging electronic quantum motion with light

Imaging the quantum motion of electrons not only in real-time, but also in real-space is essential to understand for example bond breaking and formation in molecules, and charge migration in peptides and biological systems. Time-resolved imaging interrogates the unfolding electronic motion in such systems. We find that scattering patterns, obtained by X-ray time-resolved imaging from an electronic wavepacket, encode spatial and temporal correlations that deviate substantially from the common notion of the instantaneous electronic density as the key quantity being probed. Surprisingly, the patterns provide an unusually visual manifestation of the quantum nature of light. This quantum nature becomes central only for non-stationary electronic states and has profound consequences for time-resolved imaging.     

Boundary of quantum evolution under decoherence

Relaxation effects impose fundamental limitations on our ability to coherently control quantum mechanical phenomena. In this article, we use principles of optimal control theory to establish physical limits on how closely a quantum mechanical system can be steered to a desired target state in the presence of relaxation. In particular, we explicitly compute the maximum amplitude of coherence or polarization that can be transferred between coupled heteronuclear spins in large molecules at high magnetic fields in the presence of relaxation. Very general decoherence mechanisms that include cross-correlated relaxation have been included in our analysis. We give analytical characterization for the pulse sequences (control laws) that achieve these physical limits and provide supporting experimental evidence. Exploitation of cross-correlation effects has recently led to the development of powerful methods in NMR spectroscopy to study very large biomolecules in solution. For two heteronuclear spins, we demonstrate with experiments that cross-correlated relaxation optimized pulse (CROP) sequences provide significant gains over the state-of-the-art methods. It is shown that despite large relaxation rates, coherence can be transferred between coupled spins without any loss in special cases where cross-correlated relaxation rates can be tuned to autocorrelated relaxation rates.     

On conservation laws in quantum mechanics

We raise fundamental questions about the very meaning of conservation laws in quantum mechanics, and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.

Conservation laws, such as those for energy, momentum, and angular momentum, are among the most fundamental laws of nature. As such, they have been intensively studied and extensively applied. First discovered in classical Newtonian mechanics, they are at the core of all subsequent physical theories, nonrelativistic and relativistic, classical and quantum. Here, we present a paradoxical situation in which such quantities are seemingly not conserved. Our results raise fundamental questions about the very meaning of conservation laws in quantum mechanics, and we argue that the standard way of defining conservation laws, while perfectly valid as far as it goes, misses essential features of nature and has to be revisited and extended.That paradoxical processes must arise in quantum mechanics in connection with conservation laws is to be expected. Indeed, on the one hand, physics is local: Causes and observable effects must be locally related, in the sense that no observations in a given space–time region can yield any information about events that take place outside its past light cone.^* On the other hand, measurable dynamical quantities are identified with eigenvalues of operators, and their corresponding eigenfunctions are not, in general, localized. Energy, for example, is a property of an entire wave function. However, the law of conservation of energy is often applied to processes in which a system with an extended wave function interacts with a local probe. How can the local probe “see” an extended wave function? What determines the change in energy of the local probe? These questions lead us to uncover quantum processes that seem, paradoxically, not to conserve energy.The present paper (which is based on a series of unpublished results, first described in refs. 3 and 4), presents the paradox and discusses various ways to think of conservation laws, but does not offer a resolution of the paradox.     

Physical adsorption of the quantum gas

The significant structure theory of quantum liquids such as hydrogen, deuterium, and helium has been applied to the physical adsorption occurring in the submonolayer region. The partition function for the two-dimensional adsorbed fluid has been derived. The properties of isothermal and isosteric heat show good agreement with the available data from the literature.     

Vertex representations of quantum affine algebras

We construct vertex representations of quantum affine algebras of ADE type, which were first introduced in greater generality by Drinfeld and Jimbo. The limiting special case of our construction is the untwisted vertex representation of affine Lie algebras of Frenkel-Kac and Segal. Our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations. In the case of the quantum affine algebra of type A, we introduce vertex operators corresponding to all the roots and determine their commutation relations. This provides an analogue of a Chevalley basis of the affine Lie algebra [unk](n) in the basic representation.     

Protoribosome by quantum kernel energy method

Experimental evidence suggests the existence of an RNA molecular prebiotic entity, called by us the “protoribosome,” which may have evolved in the RNA world before evolution of the genetic code and proteins. This vestige of the RNA world, which possesses all of the capabilities required for peptide bond formation, seems to be still functioning in the heart of all of the contemporary ribosome. Within the modern ribosome this remnant includes the peptidyl transferase center. Its highly conserved nucleotide sequence is suggestive of its robustness under diverse environmental conditions, and hence on its prebiotic origin. Its twofold pseudosymmetry suggests that this entity could have been a dimer of self-folding RNA units that formed a pocket within which two activated amino acids might be accommodated, similar to the binding mode of modern tRNA molecules that carry amino acids or peptidyl moieties. Using quantum mechanics and crystal coordinates, this work studies the question of whether the putative protoribosome has properties necessary to function as an evolutionary precursor to the modern ribosome. The quantum model used in the calculations is density functional theory–B3LYP/3–21G*, implemented using the kernel energy method to make the computations practical and efficient. It occurs that the necessary conditions that would characterize a practicable protoribosome—namely (i) energetic structural stability and (ii) energetically stable attachment to substrates—are both well satisfied.A suggestion of a molecular entity, called “protoribosome,” which may have evolved and emerged from an RNA world before a subsequent evolution into the modern protein/nucleic acid world, has been reported (1–3). In contemporary cells the ribosomes translate the genetic information (stored in the DNA) into proteins. Ribosomes are gigantic complexes, which in prokaryotes are built of some 50 proteins and three RNA chains with a total of 4,500 nucleotides. Aptly referred to as the protein factory of all living cells, the ribosome is essential to the contemporary life, and its activity may have been crucial to the formation of life itself. Structural analysis identified an internal RNA region that exists in all known structures (1–5) and has universally conserved sequence (1), which contains the site of peptide bond formation, and thus may well be that of a remaining RNA world entity. Consistent with the findings that the main ribosomal functions—namely, the decoding of the genetic code, the formation of peptide bonds, and the creation of elongating proteins—are performed by ribosomal RNA and with the universality of this region among all kingdoms of life, we proposed that this region is a remnant of a prebiotic chemical entity with catalytic capabilities, and called it the “protoribosome.” Within the otherwise asymmetric ribosome, this region has a unique fold (6) and could have been the link to the modern world (7). It is characterized by a pseudotwofold symmetry with a highly conserved nucleotide sequence and seems to possess all of the assumed prerequisites for the formation of chemical bonds. This semisymmetric object could be a dimer of self-folding RNA units that formed a pocket within which two activated amino acids, as substrates, might be accommodated.A representation of a plausible sequence for spontaneous self-assembly of a protoribosome is shown in Fig. 1 (2). Here, we put forth the use of quantum mechanics to answer the following question: Is the suggested protoribosome structure a plausible reality? One may systematically remove—that is, mathematically—all surrounding parts of the modern ribosome and use the coordinates of a central symmetric pocket for constructing a putative protoribosome. Here we apply quantum mechanics to the structure of that protoribosome. The most fundamental inquiry followed in this article is that of the energetic stability of the proposed protoribosome. This is not presently known. And obviously if the structure is not energetically stable, it is not likely to be able to act as a biological catalyst, as would be required of a protoribosome. The protoribosome contains almost 200 nucleotides, namely thousands of atoms. Ab initio quantum calculations rise in difficulty as a high power of the number of atoms in the system. Therefore, quantum calculation of the protoribosome energy is a complex computational problem. Fortunately we are in possession of a recently discovered kernel energy method (KEM) (8–24), described below, which alleviates dramatically the computational difficulty of ab initio calculations. Importantly the KEM is highly accurate, as well as computationally efficient.Open in a separate windowFig. 1.The scheme by which small, self-folded RNA molecules dimerize to form a symmetrical pocket allowing accommodation of a pair of substrates. The A-site region (Areg) and the P-site region (Preg), respectively, (Upper Left) dimerize (Upper Right) to allow substrate accommodation. Reproduced by permission from ref. 2 [Davidovich et al. (2009) Research in Microbiology 160(7):487–492]. Copyright Elsevier Masson SAS.As an example of the large size of systems that can be studied with ab initio KEM, we have applied the method to a Hartree–Fock (HF) calculation of a 33,000-atom protein (16). It is entirely feasible to treat even larger molecules within the context of KEM capabilities. Therefore, we performed an ab initio KEM study of the protoribosome and showed that its existence is quite feasible. Using KEM we address the question of whether the basic symmetric structure of the folded dimer pocket that constitutes the protoribosome suggested previously (4) proves to be quantum mechanically stable. If so, the next question to address is: Can it accommodate a pair of amino acids bound to a chain of a few (1–3) nucleotides, representing the tRNA 3′−end, spatially and energetically? Furthermore, such calculations should indicate the energetic preferences for the length of the nucleotide chain and its correlation to the protoribome size, ranging between 120 and 180. If both questions would be validated quantum mechanically, that would be highly suggestive of the protoribosome as an actual remnant from the RNA world still functioning in the chemistry of life, in the modern DNA/RNA/protein world.