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- Electron Thermalization in Metallic Islands Probed by Coulomb Blockade Thermometry
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The glass itself expands and contracts and leading to under and over reading of temperatures.
Electron Thermalization in Metallic Islands Probed by Coulomb Blockade Thermometry
High-precision low-temperature thermometry is a challenge for experimental quantum physics and quantum sensing. Here we consider a thermometer modeled by a dynamically-controlled multilevel quantum probe in contact with a bath. Dynamical control in the form of periodic modulation of the energy-level spacings of the quantum probe can dramatically increase the maximum accuracy bound of low-temperatures estimation, by maximizing the relevant quantum Fisher information.
As opposed to the diverging relative error bound at low temperatures in conventional quantum thermometry, periodic modulation of the probe allows for low-temperature thermometry with temperature-independent relative error bound.
The proposed approach may find diverse applications related to precise probing of the temperature of many-body quantum systems in condensed matter and ultracold gases, as well as in different branches of quantum metrology beyond thermometry, for example in precise probing of different Hamiltonian parameters in many-body quantum critical systems. Precise probing of quantum systems is one of the keys to progress in diverse quantum technologies, including quantum metrology 1 , 2 , 3 , quantum information processing QIP 4 , and quantum many-body manipulations 5.
The maximum amount of information obtained on a parameter of a quantum system is quantified by the quantum Fisher information QFI , which depends on the extent to which the state of the system changes for an infinitesimal change in the estimated parameter 6 , 7 , 8 , 9 , Ways to increase the QFI, thereby increasing the precision bound of parameter estimation, are therefore recognized to be of immense importance 11 , Recent works have studied QFI for demonstrating the criticality of environmental bath information 13 , QFI enhancement in the presence of strong coupling 14 or by dynamically controlled quantum probes 9 , and the application of quantum thermal machines to quantum thermometry Here, we propose the synthesis of two concepts: quantum thermometry 8 , 12 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 and temporally periodic dynamical control that has been originally developed for decoherence suppression in QIP 23 , 24 , 25 , 26 , We show that such control can strongly increase the QFI that determines the precision bound of temperature measurement, particularly at temperatures approaching absolute zero.
Accordingly, such control may boost the ultralow-temperature precision bound of diverse thermometers, e. Alternatively, dynamical control may allow theese thermometers to accurately estimate a broad range of temperatures.
An example of our dynamically controlled quantum thermometer DCQT is a quantum wavepacket trapped in a potential and subjected to periodic modulation, while it is immersed in a thermal bath Fig.
Measurements of the wavepacket after it has reached a steady-state provide information about the bath temperature. Another example of a DCQT is the internal state of either a two-level or a multilevel system coupled to a spin-chain bath 32 , The level separation is periodically modulated by a control field e. Many of the advanced thermometers 14 , 15 , 16 , 28 , 29 , 30 , 31 may be adapted to such dynamically controlled operation.
The frequency of the trap is periodically varied black arrow by modulating the length of the cavity, or the amplitudes of the lasers forming the trap. The wavepacket is in contact with a phonon bath, and relaxes to the corresponding thermal steady-state at long times.
For a single-mode DCQT, we consider a generic, periodic, diagonal frequency modulation:. We can assess the effectiveness of our measuring scheme from the QFI as a function of the system and bath parameters.
It obeys the relation 6 , 9 , In particular, it is exemplified below using a sinusoidal modulation characterized by a single, or few frequencies. On the other hand, a different modulation would enable the same thermometer to measure temperatures with higher accuracy bound or, equivalently, larger QFI , but at the expense of the changing of the temperature range over which the DCQT can measure accurately.
Therefore, in contrast to thermometry in the absence of any control, DCQT gives us the possibility of tuning the range and accuracy bound of measurable temperatures. As a generic example of DCQT, we consider the probe to be a periodically modulated harmonic oscillator. We first consider the thermometry of a broad class of bath spectral-response functions 18 , The extension of our results to the case of a generic bath spectrum is presented below.
Below we choose the control scheme Eq. The corresponding QFI is given by. Condition 14 yields the maximum possible advantage offered by our control scheme close to the absolute zero i. We have thus obtained a remarkable result: the advantage offered by our control scheme, expressed by the bound Eq. Regime 15 ensures that the dynamics is Markovian see Methods.
Conditions 14 , 15 allow us to achieve thermometry with a high-precision bound for very low temperatures; viz. Following this initial pulse, one can periodically modulate the system, as presented here.
In this case the QFI, albeit not maximal, would be still significantly higher than that of the same thermometer in the absence of dynamical control. The advantage offered by DCQT is not restricted to the specific bath spectra considered above. In fact, the key result of significant improvement in low-temperature thermometry by dynamical control holds for arbitrary bath spectra satisfying the Kubo—Martin—Schwinger condition see Eq.
Choosing a control scheme satisfying see Eq. The above result Eq. The lowest temperature for which Eq. As in the case of sub-Ohmic NFBS, condition 19 ensures that the thermalization time is long enough for the secular approximation to remain valid.
This in turn results in a relative error bound. The maximum QFI in this case is obtained by modulations satisfying the optimal condition. Condition 21 shows that one needs to tailor the control scheme to the bath spectrum at hand. We have shown that by subjecting a generic quantum thermometer to an appropriate dynamical control, we may increase its maximum accuracy bound of measuring a chosen range of temperatures. It falls in the category of secondary thermometers.
Namely, dynamical control allows us to perform low-temperature thermometry with temperature-independent relative error bound, at temperatures above the bound set by Eq. Our proposed control scheme can be tailored according to the bath spectra at hand, in order to maximize the QFI, and determine the number of its peaks and sensitivity ranges, thus making it highly versatile.
The dynamical control studied here has already been realized experimentally, in the context of probing the system-environment coupling spectrum In a two-level NV-center thermometer 51 , 52 , spin readout using photoluminescence may enable us to measure the N—V temperature.
These errors, which depend on the experimental apparatus, may prevent us from reaching the fundamental theoretical bound of Eq. DCQT can be highly advantageous for thermometry of diverse baths realized by many-body quantum systems in condensed matter and ultracold atomic gases, with spectra and modulations satisfying the optimal condition Eq.
We thereby open new avenues for the study of cavity quantum electrodynamics and quantum information processing 4 at extremely low temperatures. The caveat noted above is that although the dynamical control can be expected to significantly reduce the error in low-temperature thermometry, however, experimental errors may preclude the attainment of theoretical bound of temperature resolution.
An intriguing application of the proposed DCQT concerns experimentally studying the third law of thermodynamics for low-temperature many-body quantum systems, in the sense of understanding the scaling of the cooling rate with temperature 55 , 56 , In view of predictions that the cooling rate does not vanish as the absolute zero is approached for baths with anomalous dispersion, such as magnon ferromagnetic spin chains 58 , high-precision low-temperature thermometry is imperative.
High-precision multi-mode thermometry over a wide range of temperatures using our DCQT can verify whether different modes have different temperatures, and thus avoid thermalization. The versatility of our DCQT can be well-suited for simultaneous multi-mode probing of a bath with high-accuracy bound, which can be especially useful for nanometer-scale thermometry in biological systems 64 , The recently discussed need for thermometers with vanishing energy gaps for measuring low-temperatures in many-body quantum systems 17 and in strongly coupled quantum systems 18 , suggests the importance of control schemes capable of tuning the gap of a quantum thermometer to our advantage.
Application of our control scheme to thermometers modeled by many-body quantum systems, or to thermometers coupled strongly to the bath, in order to achieve high-precision low-temperature thermometry, is an interesting question, which we aim to address in the future. The control scheme presented above may have diverse applications in quantum metrology beyond thermometry.
For example, periodic modulation of energy levels in many-body quantum critical systems may be applicable to precise probing of inter-particle coupling strengths in these systems The system is coupled to the bath mode through the interaction Hamiltonian. In case the above assumption is violated, leading to non-Markovian dynamics 26 , 37 , the DCQT may be useful for revealing the absence of thermalization.
We sketch below the derivation of the master equation describing the thermalization of a system under periodic control. We refer to refs. For reviews on periodically driven open quantum systems, see refs. The time evolution operator for the periodic Hamiltonian Eq. Under the standard Born—Markov approximation in the weak thermometer-bath coupling limit, we arrive at the master equation The Kubo—Martin—Schwinger condition must be imposed, i. Giovannetti, V. Advances in quantum metrology. Braun, D.
Quantum-enhanced measurements without entanglement. Kurizki, G. Quantum technologies with hybrid systems. Natl Acad. Hauke, P. Measuring multipartite entanglement through dynamic susceptibilities.
Strobel, H. Fisher information and entanglement of non-gaussian spin states. Science , Paris, M. Quantum estimation for quantum technology. Braunstein, S. Statistical distance and the geometry of quantum states.
Correa, L. Individual quantum probes for optimal thermometry. Zwick, A. Maximizing information on the environment by dynamically controlled qubit probes.
Pasquale, A. Local quantum thermal susceptibility. Gefen, T.
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We describe a systematic series of experiments on thermalization of electrons in lithographic metallic thin films at millikelvin temperatures using Coulomb blockade thermometry CBT. Joule dissipation due to biasing of the CBT sensor tends to drive the electron system into non-equilibrium. Under all experimental conditions tested, the electron-electron relaxation is fast enough to ensure thermal electron distribution, which is also in agreement with the theoretical arguments we present. On the other hand, poor electron-phonon relaxation plays a dominant role in lifting the electron temperature above that of the bath. From a comparison of the results with the theoretical current-voltage characteristics of the thermometers we precisely determine the electron-phonon coupling constant for the common metals used. Our experiments show that it is a formidable task to attain thermal equilibrium with the bath using single-electron devices under non-zero bias conditions at 20—50 mK temperatures that are typically encountered in experiments. The conclusion concerning Coulomb blockade thermometry is more optimistic and two-fold: 1 One can now correct the errors due to bias heating in a satisfactory manner based on known material properties and the size of the metal films in the sensor.
Sensitive probing of temperature variations on nanometer scales represents an outstanding challenge in many areas of modern science and technology 1. In particular, a thermometer capable of sub-degree temperature resolution over a large range of temperatures as well as integration within a living system could provide a powerful new tool for many areas of biological, physical and chemical research; possibilities range from the temperature-induced control of gene expression 2 — 5 and tumor metabolism 6 to the cell-selective treatment of disease 7 , 8 and the study of heat dissipation in integrated circuits 1. By combining local light-induced heat sources with sensitive nanoscale thermometry, it may also be possible to engineer biological processes at the sub-cellular level 2 — 5. Here, we demonstrate a new approach to nanoscale thermometry that utilizes coherent manipulation of the electronic spin associated with nitrogen-vacancy NV color centers in diamond. We show the ability to detect temperature variations down to 1. Using NV centers in diamond nanocrystals nanodiamonds, NDs , we directly measure the local thermal environment at length scales down to nm. Finally, by introducing both nanodiamonds and gold nanoparticles into a single human embryonic fibroblast, we demonstrate temperature-gradient control and mapping at the sub-cellular level, enabling unique potential applications in life sciences.
A thermometer is a device that measures temperature or a temperature gradient the degree of hotness or coldness of an object. A thermometer has two important elements: 1 a temperature sensor e. Thermometers are widely used in technology and industry to monitor processes, in meteorology , in medicine, and in scientific research. Some of the principles of the thermometer were known to Greek philosophers of two thousand years ago. As Henry Carrington Bolton noted, the thermometer's "development from a crude toy to an instrument of precision occupied more than a century, and its early history is encumbered with erroneous statements that have been reiterated with such dogmatism that they have received the false stamp of authority. He invented the mercury-in-glass thermometer first widely used, accurate, practical thermometer   and Fahrenheit scale first standardized temperature scale to be widely used. While an individual thermometer is able to measure degrees of hotness, the readings on two thermometers cannot be compared unless they conform to an agreed scale.
Temperature is one of the most widely measured physical quantities. As a notion, it is as old as civilization itself. Yet, scientifically, the meaning and conceptual generality of temperature only fully emerged after intense efforts were made to precisely measure it starting from the 18th century on [ 1 ]. That work culminated in the discovery of the absolute temperature scale and revealed the fundamental status temperature has in thermodynamics. Advances in such measurements have forced a reassessment of basic thermodynamic quantities [ 2 ].
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Celsius, or centigrade, is a scale and unit of measurement for temperature. It is one of the most commonly used temperature units. Celsius, also known as centigrade, is a scale to measure temperature.
Heat is familiar to all of us. We can feel heat entering our bodies from the summer Sun or from hot coffee or tea after a winter stroll. We can also feel heat leaving our bodies as we feel the chill of night or the cooling effect of sweat after exercise.
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