Quantum Clock Research Articles

Demonstration of the mass-producible feature of a Cs vapor microcell technology for miniature atomic clocks

We report on the characterization of Cs vapor microcells based on pill dispensers and fabricated in a MEMS foundry according to a process compatible with mass-production. More than three quarters of cells from 6-inch wafers are successfully filled with Cs vapor. Various cells of a given wafer have been characterized using coherent population trapping (CPT) spectroscopy, demonstrating similar buffer gas (Ne) pressure with a standard deviation of about 2.5% and CPT resonances with similar linewidth and contrast properties. In addition, frequency drifts mainly attributed to cell inner atmosphere variations have been investigated onto several cells over 250–500 h measurements. The corresponding contribution at 1 day averaging time to a clock fractional frequency stability is estimated to be about 10−11 or lower. In a last section, the fractional frequency stability of a clock prototype using one fabricated Cs-Ne microcell is measured to be 2.5 × 10−11 τ−1/2 up to 200 s averaging time and better than 2 × 10−11 at 105 s. The clock frequency stability is mainly limited at short-term by the frequency-to-amplitude (FM-AM) noise conversion process and the laser amplitude (AM) noise. The mid-term stability is mainly affected by temperature-induced light shift effects. These performances tend to demonstrate that this vapor cell technology, compatible with mass-production, is suitable for miniature quantum clocks or sensors.

How Long Does a Quantum Particle or Wave Stay in a Given Region of Space?

The delay time associated with a scattering process is one of the most important dynamical aspects in quantum mechanics. A common measure of this is the Wigner delay time based on the group velocity description of a wave packet, which may easily indicate superluminal or even negative times of interaction that are unacceptable. Many other measures such as dwell times have been proposed, but also suffer from serious deficiencies, particularly for evanescent waves. One important way of realising timescales that are causally connected to the spatial region of interest relies on utilising the dynamical evolution of extra degrees of freedom, called quantum clocks, such as the precession of the spin of an electron in an applied magnetic field or the coherent decay or growth of light in an absorptive or amplifying medium placed within the region of interest. Here we provide a review the several approaches developed to answer the basic question “how much time does a quantum particle (or wave) spend in a specified region of space?” While a unique answer still evades us, important progress has been made in understanding the timescales and obtaining positive definite times of interaction by noting that all such clocks are affected by spurious scattering concomitant with the very clock potentials, however, weak they be, and by eliminating the spurious scattering.

The cosmological constant problem and quantum spacetime reference frame

We generalize the idea of quantum clock time to quantum spacetime reference frame via physical realization of a reference system by quantum rulers and clocks. Omitting the internal degrees of freedom (such as spins) of the physical rulers and clocks, only considering their metric properties, the spacetime reference frame is described by a bosonic nonlinear sigma model (NLSM). We study the quantum behavior of the system under approximations, and obtain (1) a cosmological constant valued [Formula: see text] ([Formula: see text] the critical density at near current epoch) which is very close to the observations; (2) an effective Einstein–Hilbert term; (3) the ratio of variance to mean-squared of spacetime interval tends to a universal constant [Formula: see text] in the infrared region. This effect is testable by observing a linear dependence between the inherent quantum variance and mean-squared of the redshifts from cosmic distant spectral lines. The proportionality is expected to be the observed percentage of the dark energy. The equivalence principle is also generalized to the quantum level.

Quantum Measurements, Energy Conservation and Quantum Clocks

AbstractA spin chain extending from Alice to Bob with nearest neighbors interactions, initially in its ground state, is considered. Assuming that Bob measures the last spin of the chain, the energy of the spin chain has to increase, at least on average, due to the measurement disturbance. Presumably, the energy is provided by Bob's measurement apparatus. Assuming that, simultaneously to Bob's measurement, Alice measures the first spin, it is shown that either energy is not conserved, – implausible – or the projection postulate doesn't apply, and that there is signalling. An explicit measurement model shows that energy is conserved (as expected), but that the spin chain energy increase is not provided by the measurement apparatus(es), that the projection postulate is not always valid – illustrating the Wigner–Araki–Yanase (WAY) theorem – and that there is signalling, indeed. The signalling is due to the non‐local interaction Hamiltonian. This raises the question of whether a suitable quantum‐information‐inspired model of such non‐local Hamiltonians can be developed.

Quantum stopwatch: how to store time in a quantum memory.

Quantum mechanics imposes a fundamental trade-off between the accuracy of time measurements and the size of the systems used as clocks. When the measurements of different time intervals are combined, the errors due to the finite clock size accumulate, resulting in an overall inaccuracy that grows with the complexity of the set-up. Here, we introduce a method that, in principle, eludes the accumulation of errors by coherently transferring information from a quantum clock to a quantum memory of the smallest possible size. Our method could be used to measure the total duration of a sequence of events with enhanced accuracy, and to reduce the amount of quantum communication needed to stabilize clocks in a quantum network.

Clock-Work Trade-Off Relation for Coherence in Quantum Thermodynamics.

In thermodynamics, quantum coherences-superpositions between energy eigenstates-behave in distinctly nonclassical ways. Here we describe how thermodynamic coherence splits into two kinds-"internal" coherence that admits an energetic value in terms of thermodynamic work, and "external" coherence that does not have energetic value, but instead corresponds to the functioning of the system as a quantum clock. For the latter form of coherence, we provide dynamical constraints that relate to quantum metrology and macroscopicity, while for the former, we show that quantum states exist that have finite internal coherence yet with zero deterministic work value. Finally, under minimal thermodynamic assumptions, we establish a clock-work trade-off relation between these two types of coherences. This can be viewed as a form of time-energy conjugate relation within quantum thermodynamics that bounds the total maximum of clock and work resources for a given system.

Work production of quantum rotor engines

We study the mechanical performance of quantum rotor heat engines in terms of common notions of work using two prototypical models: a mill driven by the heat flow from a hot to a cold mode, and a piston driven by the alternate heating and cooling of a single working mode. We evaluate the extractable work in terms of ergotropy, the kinetic energy associated to net directed rotation, as well as the intrinsic work based on the exerted torque under autonomous operation, and we compare them to the energy output for the case of an external dissipative load and for externally driven engine cycles. Our results connect work definitions from both physical and information-theoretical perspectives. In particular, we find that apart from signatures of angular momentum quantization, the ergotropy is consistent with the intuitive notion of work in the form of net directed motion. It also agrees with the energy output to an external load or agent under optimal conditions. This sets forth a consistent thermodynamical description of rotating quantum motors, flywheels, and clocks.

Zn clock models and chains of so(n)2 non-Abelian anyons: symmetries, integrable points and low energy properties

We study two families of quantum models which have been used previously to investigate the effect of topological symmetries in one-dimensional correlated matter. Various striking similarities are observed between certain quantum clock models, spin chains generalizing the Ising model, and chains of non-Abelian anyons constructed from the so(n)2 fusion category for odd n, both subject to periodic boundary conditions. In spite of the differences between these two types of quantum chains, e.g. their Hilbert spaces being spanned by tensor products of local spin states or fusion paths of anyons, the symmetries of the lattice models are shown to be closely related. Furthermore, under a suitable mapping between the parameters describing the interaction between spins and anyons the respective Hamiltonians share part of their energy spectrum (although their degeneracies may differ). This spin-anyon correspondence can be extended by fine-tuning of the coupling constants leading to exactly solvable models. We show that the algebraic structures underlying the integrability of the clock models and the anyon chain are the same. For n = 3,5,7 we perform an extensive finite size study—both numerical and based on the exact solution—of these models to map out their ground state phase diagram and to identify the effective field theories describing their low energy behaviour. We observe that the continuum limit at the integrable points can be described by rational conformal field theories with extended symmetry algebras which can be related to the discrete ones of the lattice models.

Sympathetic sideband cooling of a 40Ca+–27Al+ pair toward a quantum logic clock

We achieve the sympathetic sideband cooling of a 40Ca+–27Al+ pair. Both axial modes of the two-ion chain are cooled simultaneously to near the ground state of the motion. The center of mass mode is cooled to an average phonon number of 0.052(9), and the breathing mode is cooled to 0.035(6). The heating rates of both a single 40Ca+ and the 40Ca+–27Al+ pair are measured and compared. This work is a fundamental step toward the implementation of a 40Ca+–27Al+ quantum logic clock.

Universal interrogation protocol with zero probe-field-induced frequency shift for quantum clocks and high-accuracy spectroscopy

Optical clock interrogation protocols, based on laser-pulse spectroscopy, suffer from probe-induced frequency shifts and their variations induced by laser power. The original hyper-Ramsey probing scheme, which was proposed to alleviate those issues, does not fully eliminate the shift, especially when decoherence and relaxation by spontaneous emission or collisions are present. We propose to solve the fundamental problem of frequency shifts induced by the laser probe by deriving the exact canonical form of a multipulse generalized hyper-Ramsey resonance, including decoherence and relaxation. We present a universal interrogation protocol based on composite laser-pulse spectroscopy with phase modulation eliminating probe-induced frequency shifts at all orders in the presence of various dissipative processes. Unlike frequency shift extrapolation based methods, a universal interrogation protocol based on $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/4$ and $\ifmmode\pm\else\textpm\fi{}3\ensuremath{\pi}/4$ phase-modulated resonances is proposed which does not compromise the stability of the optical clock while maintaining an ultrarobust error signal gradient in the presence of substantial uncompensated ac Stark shifts. Such a scheme can be implemented in two flavors: either by inverting clock state initialization or by pulse order reversal even without a perfect quantum state initialization. This universal interrogation protocol can be applied to atomic, molecular, and nuclear frequency metrology, mass spectrometry, and the field of precision spectroscopy. It might be designed using magic-wave-induced transitions, two-photon excitation, and magnetically induced spectroscopy or it might even be implemented with quantum logic gate circuit and qubit entanglement.

The Thermodynamic Cost of Measuring Time

A simple model of an autonomous quantum clock yields a quantitative connection between the clock’s thermodynamic cost and its accuracy and resolution.

Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?

Time remains one of the least well understood concepts in physics, most notably in quantum mechanics. A central goal is to find the fundamental limits of measuring time. One of the main obstacles is the fact that time is not an observable and thus has to be measured indirectly. Here we explore these questions by introducing a model of time measurements that is complete and autonomous. Specifically, our autonomous quantum clock consists of a system out of thermal equilibrium --- a prerequisite for any system to function as a clock --- powered by minimal resources, namely two thermal baths at different temperatures. Through a detailed analysis of this specific clock model, we find that the laws of thermodynamics dictate a trade-off between the amount of dissipated heat and the clock's performance in terms of its accuracy and resolution. Our results furthermore imply that a fundamental entropy production is associated with the operation of any autonomous quantum clock, assuming that quantum machines cannot achieve perfect efficiency at finite power. More generally, autonomous clocks provide a natural framework for the exploration of fundamental questions about time in quantum theory and beyond.

A method of relativistic synchronization of moving atomic clocks and experimental verification thereof

We present results of the experiment on compensation of a relativistic shift in the timescale of a transportable hydrogen quantum clock with 3 × 10–15 instability over its motion path. Calculation of the relativistic shift of the onboard timescale of the moving clock was based on continuous measurements of its current coordinates and velocity over the path with the aid of GLONASS/GPS navigation equipment. The experimental error of the proposed method does not exceed ±1 ps.

Entanglement of quantum clocks through gravity

In general relativity, the picture of space-time assigns an ideal clock to each world line. Being ideal, gravitational effects due to these clocks are ignored and the flow of time according to one clock is not affected by the presence of clocks along nearby world lines. However, if time is defined operationally, as a pointer position of a physical clock that obeys the principles of general relativity and quantum mechanics, such a picture is, at most, a convenient fiction. Specifically, we show that the general relativistic mass-energy equivalence implies gravitational interaction between the clocks, whereas the quantum mechanical superposition of energy eigenstates leads to a nonfixed metric background. Based only on the assumption that both principles hold in this situation, we show that the clocks necessarily get entangled through time dilation effect, which eventually leads to a loss of coherence of a single clock. Hence, the time as measured by a single clock is not well defined. However, the general relativistic notion of time is recovered in the classical limit of clocks.

Bacteria as Quantum Clocks

In this study we have investigated the biological application of Wigner’s inequalities for smallest quantum clock. We have shown that the mass, size and doubling time of bacteria satisfied the Wigner’s inequalities for quantum clock. Data on 17 bacteria with mass 1×10-17 -1×10-15 kg, size 0.3-50µm and doubling time 1×103-1×105 seconds confirmed the hypothesis of Pesi? that possibly the living bacteria appear to be the smallest quantum clocks in the Nature.

Measurement of the difference in the Earth’s gravitational potentials with the help of a transportable quantum clock

The results of the Russia’s first ground-based experiment for determination of the difference in the Earth’s gravitational potentials on the basis of the measurement of the gravitational effect of the time delay with the help of a high-stability transportable atomic clock are provided. The reference atomic clock was placed in Moscow oblast, and a transportable quantum clock with an instability of 3 × 10–15 was placed in the Caucasus Mountains, with a difference in height of the clocks of 1804 m. The measured difference in the gravitational potentials between the positions of the two quantum clocks was (182.0 ± 3.1)102m2s-2 at a relative measurement error of no more than 1.7%.

Measurement of many-body chaos using a quantum clock

There has been recent progress in understanding chaotic features in many-body quantum systems. Motivated by the scrambling of information in black holes, it has been suggested that the time dependence of out-of-time-ordered (OTO) correlation functions such as $\langle O_2(t) O_1(0) O_2(t) O_1(0) \rangle $ is a faithful measure of quantum chaos. Experimentally, these correlators are challenging to access since they apparently require access to both forward and backward time evolution with the system Hamiltonian. Here, we propose a protocol to measure such OTO correlators using an ancilla which controls the direction of time. Specifically, by coupling the state of ancilla to the system Hamiltonian of interest, we can emulate the forward and backward time propagation, where the ancilla plays the role of a 'quantum clock'. Within this scheme, the continuous evolution of the entire system (the system of interest and the ancilla) is governed by a time-independent Hamiltonian. Our protocol is immune to errors that could occur when the direction of time evolution is externally controlled by a classical switch.

Sympathetic Cooling of 40Ca+−27Al+ Ion Pair Crystal in a Linear Paul Trap**Supported by the National High Technology Research and Development Program of China under Grant No 2012AA120701, and the National Natural Science Foundation of China under Grant No 11174326.

The 27Al+ ion optical clock is one of the most attractive optical clocks due to its own advantages such as low black-body radiation shift at room temperature and insensitivity to the magnetic drift. However, it cannot be laser-cooled directly in the absence of 167nm laser to date. This problem can be solved by sympathetic cooling. In this work, a linear Paul trap is used to trap both 40Ca+ and 27 Al+ ions simultaneously, and a single Doppler-cooled40 Ca+ ion is employed to sympathetically cool a single 27Al+ ion. Thus a ‘bright-dark’ two-ion crystal has been successfully synthesized. The temperature of the crystal has been estimated to be about 7 mK by measuring the ratio of carrier and sideband spectral intensities. Finally, the dark ion is proved to be an 27Al+ ion by precise measuring of the ion crystal's secular motion frequency, which means that it is a great step for our 27Al+ quantum logic clock.

Synchronizing quantum and classical clocks made of quantum particles

We demonstrate that the quantum corrections to the classical arrival time for a quantum object in a potential free region of space, as computed by Galapon [Phys. Rev. A {\bf 80}, 030102(R) (2009)], can be eliminated up to a given order of $\hbar$ by choosing an appropriate position-dependent phase for the object's wavefunction. This then implies that we can make the quantum arrival time of the object as close as possible to its corresponding classical arrival time, allowing us to synchronize a classical and quantum clock which tells time using the classical and quantum arrival time of the object, respectively. We provide an example for synchronizing such a clock by making use of a quantum object with a position-dependent phase imprinted on the object's initial wavefunction with the use of an impulsive potential.

Single electron relativistic clock interferometer

Although time is one of the fundamental notions in physics, it does not have a unique description. In quantum theory time is a parameter ordering the succession of the probability amplitudes of a quantum system, while according to relativity theory each system experiences in general a different proper time, depending on the system's world line, due to time dilation. It is therefore of fundamental interest to test the notion of time in the regime where both quantum and relativistic effects play a role, for example, when different amplitudes of a single quantum clock experience different magnitudes of time dilation. Here we propose a realization of such an experiment with a single electron in a Penning trap. The clock can be implemented in the electronic spin precession and its time dilation then depends on the radial (cyclotron) state of the electron. We show that coherent manipulation and detection of the electron can be achieved already with present day technology. A single electron in a Penning trap is a technologically ready platform where the notion of time can be probed in a hitherto untested regime, where it requires a relativistic as well as quantum description.