High-Speed Intensity Stabilization of Multiple Controller Beams by Continuous Integration Filter

Abstract

Aspects of the present disclosure describe techniques for fast stabilization of multiple controller beams using continuous integral filters. For example, a method for stabilizing the intensity of a laser beam (e.g., an ion controller beam) in a trapped ion system is described, the method including applying a linear array of laser beams to each ion in a linear array of ions in the trap; performing parallel measurements on the ions in response to the application of the laser beams, the parallel measurements including multiple separate measurements for each of the ions to identify fluctuations in the intensity of each laser beam at each ion; and adjusting the intensity of one or more of the laser beams in response to the fluctuations identified by the parallel measurements. Corresponding systems for stabilizing the intensity of a laser beam in a trapped ion system are also described.

Description

Technical Field

[0001] (Cross - reference to Related Applications) This application claims priority to U.S. Non - Provisional Patent Application No. 16 / 995,642, titled "Fast Intensity Stabilization of Multiple Controller Beams with Continuous Integrating Filter", filed on August 17, 2020, and U.S. Provisional Patent Application No. 62 / 888,668, titled "Fast Intensity Stabilization of Multiple Controller Beams with Continuous Integrating Filter", filed on August 19, 2019, the contents of both applications are hereby incorporated by reference in their entirety.

[0002] Aspects of the present disclosure generally relate to the stabilization of laser beam intensity, and more specifically, to the fast stabilization of multiple controller beams with continuous integration.

Background Art

[0003] In a trapped - ion quantum computer or a quantum information processing (QIP) system, a linear array of laser beams is focused downward onto a linear array of ions. Each laser beam is strongly focused onto its respective ion to provide some control of that ion. The diameter of the laser beam can be from 1 micrometer (μm) to 1.5 μm at the ion position, and the distance between ions can be about 5 μm.

[0004] For a quantum computer or QIP system to operate properly, it is important that the intensity of the light of each laser beam on each ion be constant. The reasons for changing the intensity of the laser beam are various. For example, the laser beam or the ion may move, causing the intensity at the ion position to decrease, or the power of the laser beam itself may change. In some cases, due to changes in the trap holding the ions, the ions may move to spots with different intensities of the laser beam. If the laser beam or the ion changes in units of tens to hundreds of nanometers, it may seriously affect the performance of the system. Due to changes in the air pressure applied to the laser beam, destructive interference may occur at the focus (the narrowest part of the laser beam, ideally the position of the ion), and the intensity of the laser beam may decrease. Therefore, it is important to monitor and stabilize the laser beam intensity over time.

[0005] One solution is to monitor the power of the laser beam with a photodiode, feedback the measured value, and adjust the amplitude of the radio frequency (RF) signal applied to the acousto-optic modulator (AOM) that controls the laser beam to stabilize the power of the laser beam. The intensity of the laser beam may refer to the amount of power per unit area. Therefore, the terms "intensity" and "power" regarding the laser beam may be used interchangeably in this disclosure. However, the technique of monitoring the laser power with a photodiode cannot correct the intensity changes caused by the pointing jitter of the laser beam, the ion position jitter, or the air pressure jitter. Furthermore, only stabilizing the laser power at the position of the photodiode does not consider the jitter induced in the path of the laser beam close to each ion of the laser beam after the photodiode.

[0006] Therefore, it is desirable to use a technique that can measure more accurately closer to the ion. One way to do this is to actually measure the intensity or power of the laser beam with the ion itself and use those measurements to control the intensity of the laser beam. The intensity of the laser beam can be scanned within a certain range while observing the reaction of the ion and calibrated periodically. Before selecting the optimal intensity point from the scan, it is necessary to fit the results of the scan. The laser beam intensity is set to the value of the optimal point until the next calibration is performed. This process is usually very slow, taking 30 seconds to 1 minute for one calibration, so it cannot be performed at a frequency of once every 10 minutes. There is a significant amount of noise fluctuating on a time scale faster than this 10-minute interval, and none of them can be corrected by the type of calibration process described above.

Summary of the Invention

Problems to be Solved by the Invention

[0007] There is a need for a new technique that can achieve faster stabilization or control of the intensity or power of the laser beam applied to the ion while enabling measurement to be performed in or near the ion where the laser beam is controlled.

Means for Solving the Problems

[0008] Below, a simplified summary of one or more aspects of the invention is presented to provide a basic understanding of such aspects. This summary is not an extensive overview of all contemplated aspects, nor is it intended to identify the main or important elements of all aspects or to delineate the scope of any or all aspects. Its sole purpose is to present some concepts of one or more aspects in a simplified form as a prelude to the more detailed description presented later.

[0009] In one aspect of the present disclosure in a trapped ion system, a method for stabilizing the intensity of a laser beam is described, the method comprising applying a linear array of laser beams to respective ions of a linear array of ions in a trap, and in response to the application of the laser beams, performing parallel measurements on the ions, the parallel measurements including a plurality of separate measurements for each ion to identify fluctuations in the intensity of each laser beam at each ion, and adjusting the intensity of one or more of the laser beams in response to the fluctuations identified by the parallel measurements.

[0010] In another aspect of the present disclosure, a system for stabilizing the intensity of a laser beam in a trapped ion system is described, the system comprising a laser source configured to apply a linear array of laser beams to respective ions of a linear array of ions in a trap, an imaging system configured to perform parallel measurements on the ions in response to the laser beams being applied, the parallel measurements including a plurality of separate measurements for each ion to identify fluctuations in the intensity of each laser beam at each ion, and an optical controller configured to adjust the intensity of one or more of the laser beams in response to the fluctuations identified from the parallel measurements.

[0011] To achieve the above and related objectives, one or more aspects are fully described below and are particularly composed of the features pointed out in the claims. The following description and the accompanying drawings show in detail specific exemplary features of one or more of its aspects. However, these features represent only a small part of the various ways in which the principles of the various aspects can be employed, and this description is intended to cover all such aspects and their equivalents.

Brief Description of the Drawings

[0012] The disclosed aspects will be described below in conjunction with the accompanying drawings. These drawings are provided for purposes of illustration, not to limit the disclosed aspects. Like reference numerals indicate like elements.

[0013]

Figure 1

Figure 2A

Figure 2B

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DETAILED DESCRIPTION OF THE INVENTION

[0014] The detailed description set forth below in connection with the accompanying drawings is intended as a description of various configurations and is not intended to represent the only configurations in which the concepts described herein may be implemented. The detailed description includes specific details for the purpose of providing a thorough understanding of the various concepts. However, it will be apparent to those skilled in the art that these concepts may be practiced without these specific details. In some instances, well-known components are shown in block diagram form to avoid obscuring such concepts.

[0015] The object of the present disclosure is to perform a high-speed experiment that measures the response of individual ions to each laser beam at the end of an experimental cycle (e.g., every few milliseconds) in order to stabilize the laser beam intensity as frequently as possible over time. By performing these high-speed calibration or stabilization techniques, it is possible to compensate for fluctuations on a time scale faster than the 10-minute intervals currently used for regular calibration. In some cases, measurements can be made after each of several experimental cycles. These measurement results may be used, if necessary and where appropriate, to increase or decrease the intensity of each laser beam via the RF amplitude driving the channels within an AOM (e.g., a multi-channel AOM). The system is configured to separately detect the response of all ions and there are separate AOM channels that drive all laser beams to all ions, so all laser beam intensities can be measured, calibrated, or stabilized in parallel (e.g., simultaneously, at the same time). Thus, the present disclosure enables many measurements to be made and the results of those measurements to be fed back to control the intensity on many laser beams in parallel. Further, to avoid shot noise resulting from the measurements, the present disclosure describes techniques for filtering such noise, an example of which includes using an integration filter to remove shot noise.

[0016] In a trapped ion quantum computer, a trapped ion system, or a QIP system, the ability to individually address each ion may be required to control the phase, frequency, and / or amplitude, and / or polarization, as required by the particular quantum gates or experiments implemented using the ions. Some trapped ion systems, for example, have 32 individually controlled ions, although this number can be dynamically adjusted to be a smaller number, a larger number, or in some cases, more than 100 ions. This same ability to individually address each ion can also be useful for performing independent measurements on each ion, such as the experiments proposed herein for performing fast calibration or stabilization of laser beam intensity. In a system that supports individual ion control, a single global laser beam can be applied or provided to all the ions within an ion chain (e.g., a chain or linear arrangement of ions within an ion trap, see, e.g., FIG. 1), and individual laser beams can be applied or provided to those ions for which quantum gates, experiments, or measurements are to be performed or implemented while controlling the ions. These counter-propagating optical beams are referred to as Raman optical beams or simply Raman beams, and are typically laser beams having a very high center frequency (e.g., 850 THz). However, these laser beams have a distinct frequency difference (e.g., 12.6 GHz) that generates a beat note, which is used as the frequency of the qubit. An RF signal is used to control the frequency or phase of the beat note generated by the laser beam, and the beat note is then used to drive the qubits (e.g., atomic or ion qubits) within the ion chain. As used in this disclosure, the terms “atomic ion,” “atom,” and “ion” may be used interchangeably to represent particles that are to be or are actually trapped in a trap to form a crystalline, linear, or similar arrangement or configuration.This type of control has conventionally been implemented by using a single global laser beam as a means of adjusting the frequency or phase of the beat note. However, this approach has several limitations, and therefore, a system that uses individual laser beams to control ions is preferred.

[0017] Therefore, the state of trapped ions can be controlled using Raman transitions, where the beat note of two Raman laser beams can be used to coherently drive internal qubits. In a quantum computer or QIP system, when controlling individual trapped ions, an array of addressing laser beams is generated, with one laser beam generated for each trapped ion. By using a multi-channel AOM, each of the addressing laser beams in the array can be controlled to have, for example, the ability to individually correct any systematic errors collected by the trapped ions, or the ability to perform measurements on the ions. In this specification, reference is made to a multi-channel AOM for controlling individual trapped ions, but it should be understood that the present disclosure need not be so limited, and the multi-channel AOM may be implemented using a plurality of single-channel AOMs or a plurality of mini multi-channel AOMs. A number of arbitrary waveform generators (AWGs) or direct digital synthesizers (DDSs) may be used in conjunction with the multi-channel AOM, with each AWG / DDS independently controlling the channels of the AOM for each respective addressing laser beam.

[0018] In connection with the approach described above, FIG. 1 is a diagram 100 showing a plurality of trapped atomic ions 106a - 106d in a linear or one-dimensional array, such as a linear crystal 110, using a linear RF trap such as a linear RF Paul trap (the linear crystal 100 may be within a vacuum chamber not shown). The linear crystal 110 is sometimes referred to as an ion chain, for example. In the example shown in FIG. 1, the trap contains a plurality of atomic ytterbium ions (e.g., 171 Yb +It may include electrodes for trapping (ions). The techniques described herein are applicable to a wide range of atomic ions other than ytterbium ions, and thus it will be understood that the present disclosure need not be limited to the use of ytterbium ions. The number of atomic ions to be trapped is configurable, and more or fewer atomic ions may be trapped. In one example, the number of ions that can be trapped is N, where N > 1 and N is a number equal to or greater than 100, and in some embodiments, N = 32 as described above. The atoms are 171 Yb + irradiated with laser (light) radiation tuned to the resonance of, and the fluorescence of the atomic ions is imaged onto a camera. In this example, the atomic ions are separated from each other by approximately 5 micrometers (μm) as can be shown by the fluorescence. The separation of the atomic ions is determined by the balance between the externally applied confinement force and the Coulomb repulsion. The atomic ions 106a - 106d may be trapped for calculation or experiment, but the atomic ions 106a - 106d may also be used as described herein to measure each laser beam and stabilize the intensity of those laser beams.

[0019] FIG. 2A shows a diagram 200 illustrating an example of a Raman beam geometry. In diagram 200, there are individual laser beams 210 (e.g., one for each atomic ion) and a global laser beam 220 directed towards a linear crystal or chain such as linear crystal 110 having a linear array of atomic ions or qubits. The measurement techniques described herein can be used to control the intensity of laser beam 210 such that the intensity of each of the individual laser beams 210 stabilizes, at least from the perspective of each respective ion. Laser beams propagating or traveling in the same direction may be referred to as co-propagating laser beams, and laser beams propagating or traveling in different or opposite directions may be referred to as non-co-propagating or counter-propagating laser beams, respectively. Laser beams 210 (co-propagating) are laser beams focused or individually addressed to each respective atomic ion, and global laser beam 220 (counter-propagating with respect to laser beams 210 as shown) may be a global beam. This typically focuses on a wide elliptical spot covering all the ions. As used herein, the terms "laser beam", "optical beam", "beam", "laser", "optical field", and "field" may be used interchangeably. Further, the terms "atom", "atomic ion", and "ion" may also be used interchangeably.

[0020] FIG. 2B is a diagram 200b explaining an example in which fluctuations in the laser beam or ion position result in intensity fluctuations in the ion. In this example, laser beam 210 (e.g., an individual addressed laser beam) is focused at the location or position of ion 106 (solid line). The diameter 230 of the focused portion of laser beam 210 is about 1 μm to 1.5 μm. The intensity profile 240 on the right side of diagram 200b shows that ion 106 is aligned with the highest point of intensity profile 240.

[0021] A change in the position of ion 106 or a change in the position of laser beam 210 (dashed line) changes the location where ion 106 aligns with intensity profile 240, and as a result, the intensity of laser beam 210 seen by ion 106 will be lower.

[0022] The techniques described herein are used to correct for these types of fluctuations by changing the intensity of laser beam 210, where the intensity may be increased or decreased so that ion 106 sees a stable or constant laser beam intensity over time.

[0023] Figure 3 shows diagram 300, which depicts a portion of a trapped ion quantum computer or QIP system that can be used to perform measurements for the purpose of rapidly stabilizing multiple controller laser beams through continuous integration of multiple measurements. As shown in diagram 300 of Figure 3, four ions 106a - 106d are shown trapped in a linear crystal 110, although the linear crystal 110 may have fewer or more ions, and it may be possible to measure a full set or any subset of the ions to stabilize the respective laser beam intensities. Also shown in diagram 300 are multi - channel AOM 330, waveform generators 350a - 350d, imaging system 360, stabilization measurement controller 370, and multi - channel AOM controller 380. In one embodiment, AOM 330 may be a multi - channel Bragg cell having piezoelectric transducers 312a - 312d (e.g., piezoelectric transducers) arranged in parallel to locally apply acoustic waves to AOM 330 to adjust one or more characteristics (e.g., intensity) of each individual addressed laser beam 210a - 210d applied to the transducer. These laser beams 210a - 210d control aspects of the operation and measurement of ions 106a - 106d and may sometimes be referred to as controller laser beams or simply controller beams. Further, since laser beams 210a - 210d are used in a linear arrangement that matches the linear arrangement of ions 106a - 106d within linear crystal 110, they may also be referred to as a linear array of laser beams. AOM 330 may include different channels (e.g., separate transducers) for use with each of ions 106a - 106d and their respective laser beams 210a - 210d. In this example, it is shown that four channels have transducers 312a - 312d, which can be controlled by respective waveform generators 350a - 350d. Waveform generators 350a - 350d may be arbitrary waveform generators (AWG) and / or direct digital synthesizers (DDS), or other types of signal - generating devices.

[0024] During operation, in some embodiments, the global laser beam 220 may be provided to the ions 106a-106d from a first direction. The laser beams 210a-210d may be provided to illuminate some or all of the ions 106a-106d separately or individually from a second direction. During quantum operations, for example, the ions being illuminated depend on the quantum gate being implemented at the current stage of the sequence that is part of the quantum operation or algorithm. During calibration or stabilization, the ions being irradiated depend on which laser beam intensity is being measured for calibration or stabilization. In the example shown in diagram 300, the laser beam 210a may be irradiated or focused on the ion 106a, the laser beam 210b may be irradiated or focused on the ion 106b, the laser beam 210c may be irradiated or focused on the ion 106c, and the laser beam 210d may be irradiated or focused on the ion 106d. In some examples, some of the ions within the linear crystal 110 are not illuminated by any of the laser beams, and the laser beam intensity stabilization measurements may be performed in parallel using only those ions that are irradiated. In other embodiments, it may be advantageous to irradiate all of the ions within the linear crystal 110 and perform laser beam intensity stabilization measurements in parallel for all of those ions.

[0025] Diagrams 200a of FIG. 2A and 300 of FIG. 3 use a global laser beam 220 in one direction and a set of individual addressing laser beams 210 in another direction. Instead, it will also be understood that two sets, i.e., one set in one direction and one set in another direction (e.g., at different angles or on opposite sides of the trap), of individual addressing laser beams can be used.

[0026] Continuing to refer to FIG. 3, in certain embodiments, the specific characteristics of each of the laser beams 210a-210d can be individually modulated by the AOM 330. For example, the waveform generator 350a can generate and / or control an RF signal that causes an acoustic wave to be generated in one of the channels of the AOM 330 at the transducer 312a, thereby controlling the characteristics (e.g., frequency, amplitude, and / or phase) of the laser beam 210a incident on the transducer 312a. By controlling the frequency of the laser beam 210a, the difference between its frequency and the frequency of the global laser beam 220 can be utilized to generate a beat note that, in turn, controls the state of the ion 106a. The waveform generator 350a can dynamically change the RF signal to vary the characteristics of the laser beam 312a over time. For example, the amplitude of the RF signal can be used to control the intensity and / or power of the laser beam 210a applied to the ion 106a, and the waveform generator 350a can dynamically adjust or vary the amplitude of the RF signal to change the intensity based on the type of calculation or experiment being performed. This function can also be used to stabilize the intensity of the laser beam 210a.

[0027] A similar approach as described above can also be implemented with respect to the waveform generators 350b-350d, the transducers 312b-312d, and the laser beams 210b-210d. In other words, the intensity of each of the laser beams 210b-210d can be individually and dynamically controlled by controlling the amplitude of the RF signal applied to each channel (e.g., each transducer) within the AOM 330.

[0028] Also, Diagram 300 of FIG. 3 shows an imaging system 360 configured to image and analyze the effect of the interaction between the linear arrays of laser beams 210a to 210d and the respective linear arrays of ions 106a to 106d. The imaging system 360 may be used as part of the measurements performed for the high-speed stabilization of the laser beams 210a to 210d. For example, the imaging system 360 may detect the respective luminance (or the lack thereof) of the ions 106a to 106d and may be used, for example, to assign a value of "0" to ions in the dark state and a value of "1" to ions in the bright state.

[0029] Also, diagram 300 of FIG. 3 shows a stabilization measurement controller 370, which can be configured to control high-speed experiments used to measure the response of individual ions to each laser beam at the end of an experimental cycle (e.g., every few milliseconds) in order to stabilize the laser beam intensity as frequently as possible over time. The stabilization measurement controller 370 can be configured to control one or more of the scheduling of measurements (e.g., when and for how long to take measurements), the measurements to be taken (e.g., for which ions, how many measurements to take, and how to prepare the ions for measurement), and the processing of measurements (e.g., determining whether adjustments to the intensity of the laser beam are necessary). The stabilization measurement controller 370 may also be configured to generate commands for the multi-channel AOM controller 380 in order to control, if necessary, the adjustment of the RF signals generated by the waveform generators 350a - 350d, and these signals are then applied to channels in the AOM 330 to modify the intensity of the laser beams 210a - 210d. In one example, the stabilization measurement controller 370 may include an integration filter 375 configured to integrate (e.g., continuously integrate) the measurement results over time in order to obtain a more accurate representation of any fluctuations in the laser beam intensity at ions 106a - 106d. The integration filter 375 can be effectively used to set a threshold such that when too many measurements for a particular ion indicate that the laser beam intensity is lower than the desired laser beam intensity, the stabilization measurement controller 370 provides instructions to the multi-channel AOM controller 380 to increase the intensity of each laser beam by adjusting the amplitude of the RF signal of each waveform generator. Similarly, when too many measurements for a particular ion indicate that the laser beam intensity is higher than the desired laser beam intensity, the stabilization measurement controller 370 provides instructions to the multi-channel AOM controller 380 to decrease the intensity of each laser beam by adjusting the amplitude of the RF signal applied to each channel of the AOM 330.

[0030] In some embodiments, some or all of the functions of the stabilization measurement controller 370 and / or the integration filter 375 may be implemented as part of the multi-channel AOM controller 380, as parts of the imaging system 360, or as parts of both the imaging system 360 and the multi-channel AOM controller 380.

[0031] The trapped ion quantum computer or QIP system shown in diagram 300 of FIG. 3 is flexible enough not only to perform quantum calculations and experiments, but also to perform experiments or measurements for rapidly stabilizing a plurality of controller laser beams (e.g., laser beams 210a - 210d) in continuous integration. That is, this system can perform a high-speed experiment to measure the response of the linear array of ions 106a - 106d to the linear array of laser beams 210a - 210b at the end of one or more consecutive experimental cycles, and the measurement results are used to increase or decrease the laser beam intensity by adjusting the amplitude of the RF signal driving the AOM 330. The system can detect the response of each ion separately, and since there are separate channels for the AOM 330 driving the laser beam corresponding to each ion, all of the laser beam intensities can be measured in parallel and / or adjusted.

[0032] One proposed approach can proceed as follows for each that is measured (e.g., for each ion). First, each ion in the chain to be measured is prepared in the quantum state │0>. Next, the frequency is set to perform Rabi flops between the quantum states │0> and │1>, and each laser beam for each ion to be measured is turned on for a fixed time t. The on-time t for each of the laser beams is set such that, if the power or intensity of the laser beam is appropriate, each ion is driven to an equal superposition of the quantum states │0> and │1>. Next, the state of each ion is measured as either "0" or "1". Here, "1" indicates the bright state of the ion and "0" indicates the dark state of the ion. The details of this approach will be explained in more detail below.

[0033] A Rabi flopping or Rabi cycle is the periodic behavior of a two-level quantum system in the presence of an oscillating field. For example, when an atom or an ion is irradiated by a laser beam, it may periodically absorb photons from the laser beam, move to a certain energy level, and then re-emit the photons and return to another energy level. This behavior is reflected in what is called Rabi oscillation, which represents the probability that a two-level quantum system in the first quantum state (e.g., quantum state │0>) reaches the second quantum state (e.g., quantum state │1>). The duration of the Rabi cycle and its reciprocal are called the Rabi frequency of the laser beam.

[0034] Figure 4 shows a diagram 400 by way of an example of Rabi flopping, which shows the ion luminance (vertical axis) corresponding to the pulse area (horizontal axis) proportional to the intensity of the laser beam and time t. Diagram 400 is shown as a scan of the intensity of the laser beam (e.g., any one of laser beams 210a to 210d). The pulse area, i.e., the horizontal axis of diagram 400, is a parameter proportional to the intensity (I) and the fixed time t during which the laser beam is applied. The ion luminance, i.e., the vertical axis of diagram 400, can be detected by, for example, the imaging system 360 and is an oscillatory or sinusoidal behavior corresponding to sin(k·I·t). Here, k is a proportionality coefficient, and I and t are as described above.

[0035] As described above, the approach proposed for each of the measurements performed involves preparing the ion in an initial quantum state, e.g., the quantum state │0>, and setting the amount of time t for which a laser beam that drives a Rabi flopping between the quantum states │0> and │1> is applied. When the intensity is appropriately selected such that each ion is driven to an equal superposition of the │0> quantum state and the │1> quantum state, the operating point (indicated by the black dot) is at the 50% level of the ion brightness (or 0.5 on the vertical scale). This is because there will be an equal number of measurements of the bright state (“1”) and the dark state (“0”) for that ion at that intensity. When the measurements begin to show more bright states (e.g., “1” detected for the ion by the imaging system 360) than dark states (e.g., “0” detected for the ion by the imaging system 360), the operating point (black dot) moves above the curve (white dots) to have a higher ion brightness. In this case, the calibration or stabilization operation may require reducing the amplitude of the RF signal that controls the channel in the AOM330 that changes the intensity of the laser beam. When the measurements begin to show more dark states (e.g., “0” detected for the ion by the imaging system 360) than bright states (e.g., “1” detected for the ion by the imaging system 360), the operating point (black dot) moves below the curve (white dots) to have a lower ion brightness. In this case, the calibration or stabilization operation may require increasing the amplitude of the RF signal that controls the channel in the AOM330 that changes the intensity of the laser beam.

[0036] In the example shown in diagram 400, the duration for which each laser beam is turned on is set so that ions are measured at 50% of the bright state in time (e.g., │1> quantum state), as indicated by the correct intensity of the laser beam and the black dot being at the center of the ion brightness range (0.0 - 1.0). In this particular case, time t is set to drive a 9π / 2 pulse (e.g., pulse area is k·I·t), but it also operates at an odd multiple of π / 2. It is possible to set time t to drive a pulse even if it is not an odd multiple of π / 2, but using an odd multiple of π / 2 can maximize the sensitivity to intensity fluctuations. When the intensity increases or decreases, the pulse area changes in the same way, and the black dot may move up and down the sine curve representing the Rabi flopping. That is, it deviates from the case where ions are measured at 50% of the bright state at that time. For small intensity fluctuations (e.g., small changes for the black dot), this response is approximately linear. Another aspect of the flexibility provided by the proposed approach is that a large odd multiple of π / 2 for the pulse is more sensitive to intensity fluctuations than a small odd multiple of π / 2. This allows the optimal pulse to be selected and used when adjusting the intensity of the laser beam.

[0037] Since each of the measurements made on a particular ion returns either "0" or "1", these measurements may contain significant shot noise. Therefore, it is useful to pass these measurements through one (or more) integration filters to integrate several individual measurements and better identify intensity fluctuations. One embodiment for achieving shot noise reduction by continuous integration may be as follows.

[0038] As described above in connection with diagram 300 of FIG. 3, the intensity of the laser beam (e.g., laser beams 210a - 210d) may be controlled by controlling the amplitude of the RF signal that modulates each of the laser beams via the respective channel / transducer in the AOM 330. The multi - channel AOM controller 380, or a combination of the stabilization measurement controller 370 and the multi - channel AOM controller 380, may be used to provide instructions for controlling the amplitude of the RF signals generated by the waveform generators 350a - 350d.

[0039] In one embodiment, the amplitude of each RF signal may be physically determined by a 16 - bit number (or some M - bit number). This digital number may be referred to as a physical number or physical bit provided to the waveform generators 350a - 350d to generate the appropriate amplitude of the RF signal and thus the appropriate laser beam intensity. In one example, the multi - channel AOM controller 380 may provide to each of the waveform generators 350a - 350d its respective physical number (e.g., a 16 - bit number), that is, the digital number representative of the amplitude of the RF signal to be generated by each of the waveform generators 350a - 350d. The digital numbers for the amplitudes of different RF signals may be provided to the multi - channel AOM controller 380 by the stabilization measurement controller 370. However, inside the stabilization measurement controller 370, which may be implemented by a field - programmable gate array (FPGA) or an application - specific integrated circuit (ASIC), the amplitude for each RF signal and thus for each ion i may instead be determined based on a 32 - bit number (or some 2M - bit number), A i It is possible to determine based on A i In general, A i need not be limited to a 2M - bit number and may be any (M + N) - bit number (both M and N are integers). Since the 16 least - significant bits (LSB) of A i are truncated for output, they do not affect the amplitude of each RF signal (i.e., the 16 most - significant bits (MSB) of A iThe 16 least significant bits are non-physical bits). The stabilization measurement controller 370 determines, for each ion i, a second digital number B i This is the gain of the feedback loop of the measured value and is 16 bits or less. The second digital number B i In a sense, represents the result of the integral filtering performed by the integral filter 375 and is used to determine whether multiple measurements need to increase or decrease the amplitude of their respective RF signals. In one example, A i is the amplitude for ion i in the form of a 32-bit number, which can be based on the base amplitude A0 and a proportionality coefficient K of the feedback gain less than 1 (e.g., an integral gain less than 1). In the example described herein, when the second digital number B i is 16 bits, the proportionality coefficient can be equal to 1 / 16. The integration operation provides the second digital number B i In this form, S(t) is determined by a series of '1's and '0's. For example, if the measured value at time t is '1', then S(t) = 1, and if the measured value at time t is '0', then S(t) = -1. The proportionality coefficient is an adjustable parameter of the stabilization system and can be adjusted by the value of B i Since the values to be integrated are '1's and '0's, the integration is simply the sum of the values to be integrated. Overall, if the number of '1's is 16 samples or more more than the number of '0's during the integration period, the amplitude A i is adjusted in one direction, leading the intensity below the curve and back to the black point as shown in the Rabi flopping diagram of FIG. 4. Also, if the number of '0's is 16 samples or more more than the number of '1's during the integration period, the amplitude A i is adjusted in the opposite direction, leading the intensity above the curve and back to the black point as shown in the Rabi flopping diagram of FIG. 4.

[0040] In other words, after each measurement regarding ion i, if the measured value is '0' (e.g., a measurement of the dark state), the value of the digital number A i for ion i is replaced with the value of A i +B i and if the measured value is '1' (e.g., a measurement of the bright state), the digital number A of ion ii Replace the value of A i with -B i This is done to adjust the amplitude of the RF signal used to control the intensity of the laser beam used for each ion i. B i is added to the 16 LSBs of A i so that a single measurement in B i does not affect the upper 16 MSBs of A i That is, since this only affects the non-physical bits of A i it does not affect the physical amplitude. When the ion is in an equal superposition state of "0" and "1", statistically there are an equal number of "0" measurements and "1" measurements, and the physical amplitude is kept approximately constant (when B i is small enough). That is, when the number of "0" measurement values and "1" measurement values is the same, for example, the measurement values processed by the integration filter 375 are integrated to 0 or a value close to it. However, when the superposition of the ions is unbalanced (for example, when the number of "0" measurements and "1" measurements is not the same), the integration filter 375 finally pushes out the physical bits of A i in a direction to stabilize the power or intensity of each laser beam. In this case, when there are significantly more "1"s than "0"s, or significantly more "0"s than "1"s, the value of B i can become much larger than when replacing the value of A i with the value of A i +B i or the value of A i -B i This affects the physical bits of Ai and thus the physical amplitude of each RF signal.

[0041] As described above, this process can be performed in parallel for all laser beams. Further, since the measurements can be performed in parallel, these measurements and their feedback stabilization mechanisms require less than 1 millisecond (<1 ms), and thus can be interleaved between experimental cycles and executed every few milliseconds. This provides a relatively fast feedback loop that can cancel both slow drifts and fast timescale fluctuations better than what is currently achievable. Further, while the feedback stabilization mechanism described herein has been described in relation to an integral filter, it will be understood that the feedback stabilization mechanism is not so limited and can be generally applied by more complex filter functions where the integral filter is a particular implementation.

[0042] FIG. 5A is a block diagram illustrating an example of a QIP system 500 according to an aspect of the present disclosure. The QIP system 500 may also be referred to as a quantum computing system, a computer device, a trapped ion system, a trapped ion quantum computer, etc. In one aspect, the QIP system 500 may be configured to perform quantum computing and quantum experiments. Further, the QIP system 500 may be configured to perform a high-speed experiment that measures the response of individual ions to each laser beam at the end of an experimental cycle (e.g., every few milliseconds) in order to stabilize the laser beam intensity associated with each individual ion as frequently as possible over time. This stabilization procedure may be performed in parallel for multiple laser beams and the ions associated therewith.

[0043] The QIP system 500 can include a source 560 that provides a species of atoms (e.g., a flux of neutral atoms) to a chamber 550 having an ion trap 570, and the ion trap 570 traps a species of atoms that has been once ionized (e.g., photoionized) by an optical controller 520 (see, e.g., FIG. 5B). The ion trap 570 may be used to trap ions in a linear array such as the linear crystal 110 described above in connection with the diagrams 100 and 300 of FIGS. 1 and 3, respectively. The light source 530 within the optical controller 520 may include one or more laser sources (e.g., a source of optical beams or laser beams), and the laser sources can be monitored and tracked by an image processing algorithm operating in an imaging system 540 within the optical controller 520 for ionization of the species of atoms, for fluorescence of atomic ions that can be monitored and tracked, and / or for performing the optical control functions described in this disclosure in connection with laser beam intensity stabilization. The light source 530 may be configured to control and generate a linear array of laser beams such as the laser beams 210a - 210d described above in connection with the diagram 300 of FIG. 3. In one aspect, the light source 530 may be implemented separately from the optical controller 520.

[0044] The imaging system 540 may include a high - resolution imaging device (e.g., a CCD camera) for monitoring atomic ions while they are provided to the ion trap or after they are provided to the ion trap 570. In one aspect, the imaging system 540 can be implemented separately from the optical controller 520, but the use of fluorescence to detect, identify, and label atomic ions using an image processing algorithm may need to be coordinated with the optical controller 520. The imaging system 540 can be an example of the imaging system 360 described above in connection with the diagram 300 of FIG. 3. Thus, the imaging system 540 can be configured to image and analyze the effects of the interaction between a linear array of laser beams generated by the light source 530 and each linear array of ions trapped in the ion trap 570.

[0045] The QIP system 500 can also include an algorithm component 510, which can operate with other parts (not shown) of the QIP system 500 to execute a sequence of stacks or combinations of single qubit operations or multi qubit operations (e.g., two qubit operations), and quantum algorithms or quantum operations including extended quantum computing. In this way, the algorithm component 510 can provide instructions to various components (e.g., the optical controller 520) of the QIP system 500 to enable the implementation of quantum algorithms or quantum operations.

[0046] FIG. 5B shows at least a part of the optical controller 520. In this example, the optical controller 520 may include a light source 530 and an imaging system 540. As shown by the dotted line, one or both of the light source 530 and the imaging system 540 can be optionally implemented separately from but communicable with the optical controller 520.

[0047] The imaging system 540 may include a CCD 541 (or a similar imager or camera) and an image processing algorithm component 542 for processing the information captured by the CCD 541. The imaging system 540 may be used to detect the results of the measurements described herein for high-speed calibration or high-speed stabilization of the laser beam intensity. The light source 530 may include AWGs (or DDSs) 532a - 532d and lasers 534a - 534d, which may be used to control the ions within the ion trap 570. The AWGs 532a - 532d may be examples of the waveform generators 350a - 350d described above in relation to the diagram 300 of FIG. 3, and the lasers 534a - 534d may also be used to generate a linear array of the laser beams 210a - 210d shown in the same diagram 300. Also, the light source 530 may include a global modulator 535 and a global laser 536 for emitting and controlling the global Raman laser beam 220.

[0048] The optical controller 520 may include a multi-channel AOM controller 537 configured to control the operation of the multi-channel AOM 538, which respectively correspond to the above-described multi-channel AOM controller 380 and multi-channel AOM 330. The multi-channel AOM 538 may be implemented using a single AOM device having a plurality of channels, a plurality of AOM devices having a single channel, or a plurality of AOM devices having a plurality and / or single channel.

[0049] The optical controller 520 may further include a stabilization measurement controller 539 that may correspond to the above-described stabilization measurement controller 370. The stabilization measurement controller 539 may include an integration filter (not shown). The integration filter, or integrator, is a device or component whose output signal is the time integral of its input signal. That is, the integration filter accumulates the input quantity over a defined time and generates a representative output. The integration filter may be used to filter the effects of shot noise resulting from a plurality of measurements made as part of the laser beam intensity stabilization described herein. The integration filter may be an example of the above-described integration filter 375.

[0050] It will be understood that one or more of the components or sub-components of the optical controller 520 may be implemented separately from the optical controller 520. Further, one or more of the components or sub-components of the optical controller 520 may be implemented as part of one or more integrated circuits (e.g., FPGA, ASIC, central processing unit, microprocessor). In one example, the multi-channel AOM controller 537 and the stabilization measurement controller 539 may be implemented as a single component on the same integrated circuit or as separate components in different integrated circuits. When on the same integrated circuit, the multi-channel AOM controller 537 and the stabilization measurement controller 539 may use an internal 32-bit number A i and the 16 MSBs of A i are physical bits, and A iThe 16 LSBs of A are non-physical bits and i the 16 LSBs of i may be truncated for output.

[0051] Referring to FIG. 6, this is a method 600 for intensity stabilization of a laser beam in a trapped ion system. In one aspect, the functions of method 600 may be performed by one or more components of a QIP system such as a trapped ion system or QIP system 500 and its components (e.g., optical controller 520 and its components or sub-components).

[0052] At 610, method 600 includes applying a linear array of laser beams (e.g., laser beams 210a - 210d) to respective ions in a linear array of ions (e.g., ions 106a - 106d) within a trap (e.g., ion trap 570).

[0053] At 620, method 600 includes performing parallel measurements on the ions (e.g., by imaging system 360) in response to the application of the laser beams, where the parallel measurements include a plurality of separate measurements for each ion to identify the intensity fluctuations of the respective laser beams at each ion. For example, the measurements may be performed following (e.g., immediately after) the application of the laser beams.

[0054] At 630, method 600 includes adjusting the intensity of one or more of the laser beams (e.g., by stabilization measurement controller 370, multi-channel AOM controller 380, and / or waveform generators 350a - 350d) in response to the fluctuations identified from the parallel measurements.

[0055] Method 600 generally corresponds to a sequence that may apply pulsed laser beams, measure ion states in parallel, and update the laser beam output based on the measurements. Such a sequence may be repeated multiple times.

[0056] In one aspect of method 600, each of a plurality of separate measurements at a particular ion is a measurement of the response of that ion to each laser beam.

[0057] In one aspect of method 600, the step of performing parallel measurements includes performing integral filtering (e.g., by integral filter 375) of a plurality of separate measurements for each ion to identify intensity fluctuations of each laser beam.

[0058] In one aspect of method 600, the step of adjusting the intensity of one or more of the laser beams includes controlling each of the laser beams using an RF signal applied to each channel within a multi-channel AOM (e.g., multi-channel AOM 330). Controlling each of the laser beams using the RF signal includes adjusting the amplitude of the RF signal in response to identifying intensity fluctuations of each laser beam.

[0059] In another aspect of method 600, for each of a plurality of separate measurements for a particular ion, method 600 includes the step of preparing the ion in the quantum state │0>, where each laser beam has a correct intensity and, when at a frequency that drives Rabi flopping of the ion (e.g., see FIG. 4) between the quantum states │0> and │1>, is applied to that ion during a fixed period t that generates an equal superposition of the quantum states │0> and │1>, and the step of measuring the response of the ion to the application of each laser beam as either "1" or "0" (e.g., by imaging system 360), where "1" indicates the bright state of the ion and "0" indicates the dark state of the ion. The fixed period t may be set to drive a laser beam pulse that is an odd multiple of π / 2, and a larger odd multiple of π / 2 is more sensitive to intensity fluctuations than a smaller odd multiple of π / 2. Further, the fixed period t may be set to drive a laser beam pulse of 9π / 2.

[0060] In another aspect of method 600, each laser beam has a diameter of about 1 μm to 1.5 μm at the position of each ion, and the ions in the linear array of ions are separated from each other by a distance of about 5 μm.

[0061] In another aspect of method 600, method 600 may be performed at the end of an experimental cycle of a trapped ion system (e.g., QIP system 500), or may be performed during successive experimental cycles of the trapped ion system. Overall, method 600 may be performed in less than 1 millisecond, which is substantially faster than existing methods that are not performed in parallel.

[0062] In one aspect of method 600, the step of controlling each of the laser beams using an RF signal includes adjusting a first digital number A corresponding to the amplitude of the RF signal based on a second digital number B corresponding to the gain of a feedback loop of a plurality of separate measurements, in response to identifying fluctuations in intensity in each of the laser beams. The most significant bit (MSB) of the first digital number A is the bit representing the physical value of the amplitude of each RF signal.

[0063] In another aspect of method 600, each of the measurements may first include the step of preparing each ion in the chain to be measured in the quantum state │0>. Next, each laser beam for each ion to be measured is turned on for a fixed period of time t at a frequency set to drive Rabi flopping between the quantum states │0> and │1>. The period t for each laser beam to be turned on is set such that, if the power or intensity of the laser beam is appropriate, each ion is driven to an equal superposition of the quantum states │0> and │1>. Then, the state of each ion is measured as "0" or "1". Here, "1" indicates the bright state of the ion, and "0" indicates the dark state of the ion.

[0064] Generally, the techniques described herein in connection with FIGS. 1-6 above can be implemented using a system for stabilizing the intensity of a laser beam in a trapped ion system. The system includes a laser source configured to apply a linear array of laser beams to respective ions within a trap, and an imaging system configured to perform parallel measurements on the ions while the laser beams are being applied, the parallel measurements including a plurality of separate measurements for each of the ions to identify fluctuations in the intensity of each laser beam at each ion, and an optical controller configured to adjust the intensity of one or more of the laser beams in response to the fluctuations identified from the parallel measurements.

[0065] In another aspect of the system described in connection with FIGS. 1-6, each of the plurality of separate measurements for a particular ion is a measurement of the response of that ion to each laser beam. For each of the plurality of separate measurements for a particular ion, the optical controller is configured to prepare the ion in the quantum state │0>, and each laser beam is applied to that ion for a fixed period t that generates an equal superposition of the quantum states │0> and │1> when the intensity of the laser beam is correct and at a frequency that drives Rabi flopping of the ion between the quantum states │0> and │1>. The imaging system is configured to measure the response of the ion to the application of each laser beam as either "1" or "0", where "1" indicates the bright state of the ion and "0" indicates the dark state of the ion. The fixed period t may be set to drive a laser beam pulse that is an odd multiple of π / 2, and larger odd multiples of π / 2 are more sensitive to intensity fluctuations than smaller odd multiples of π / 2. The fixed period t may be set to drive a laser beam pulse of 9π / 2.

[0066] In another aspect of the system described in relation to FIGS. 1 - 6, the optical controller is further configured to perform integral filtering of a plurality of separate measurements for each ion to identify intensity fluctuations in each laser beam. The optical controller configured to adjust the intensity of one or more of the laser beams is further configured to control each of the laser beams using the RF signals applied to each channel in the multi - channel AOM. The optical controller configured to control each of the laser beams using the RF signals is further configured to adjust the amplitude of the RF signals in response to the identification of intensity fluctuations in each laser beam. The optical controller configured to control each of the laser beams using the RF signals is further configured to adjust a first digital number A corresponding to the amplitude of the RF signal by a second digital number B corresponding to the gain of the feedback loop of the plurality of separate measurement values in response to the identification of intensity fluctuations in each laser beam. The MSB of the first digital number A is the bit representing the physical value of the amplitude of each RF signal.

[0067] In another aspect of the system described in relation to FIGS. 1 - 6, each laser beam has a diameter of about 1 μm to 1.5 μm at the position of each ion, and the ions in the linear array of ions are separated from each other by a distance of about 5 μm.

[0068] In another aspect of the system described in relation to FIGS. 1 - 6, the imaging system is configured to perform parallel measurements at the end of the experimental cycle in the trap ion system. The imaging system is configured to perform parallel measurements during the experimental cycle in the trap ion system. The imaging system is configured to perform parallel measurements in less than 1 millisecond.

[0069] The foregoing description of the disclosure is provided so that those skilled in the art can make or use the disclosure. Various modifications to the disclosure will be readily apparent to those skilled in the art, and the common principles defined herein can be applied to other variations without departing from the spirit or scope of the disclosure. Further, although the elements of the described embodiments may be described or claimed in the singular, the plural is also contemplated unless the limitation to the singular is explicitly stated. Additionally, all or part of any aspect may be utilized with all or part of any other aspect, unless otherwise specified. Accordingly, the disclosure is not limited to the examples and designs described herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for stabilizing the intensity of a laser beam in a trapped ion system, comprising: applying a linear array of laser beams to respective ions of a linear array of ions in a trap; performing parallel measurements on said respective ions in response to the application of said laser beams, said parallel measurements including a plurality of separate measurements for each ion in said linear array, each of said plurality of measurements for an ion indicating either a bright state or a dark state of said ion; identifying fluctuations in the intensity of said respective laser beams at each ion based on said plurality of separate measurements for said respective ions; adjusting said intensity of one or more of said laser beams in response to the fluctuations identified by said parallel measurements; and a method comprising the steps of:

2. The method according to claim 1, wherein each of said plurality of separate measurements for a particular ion is a measurement of the response of that ion to said respective laser beam.

3. The method according to claim 1, wherein the step of performing the parallel measurements includes integrating said plurality of separate measurement values for each of said ions to identify fluctuations in the intensity of said respective laser beams.

4. For each of said plurality of separate measurements for a particular ion, preparing said ion in a quantum state │0>, wherein each laser beam is applied to that ion for a fixed period t that generates an equal superposition of the quantum states │0> and │1> when the intensity of said laser beam is correct and drives Rabi flopping of said ion between the quantum states │0> and │1>; measuring the response of said ion to the application of said respective laser beams as either "1" or "0", where "1" indicates the bright state of said ion and "0" indicates the dark state of said ion; and a method according to claim 1, comprising the steps of:

5. The method according to claim 4, wherein said fixed period t is set to drive a laser beam pulse that is an odd multiple of π / 2, and a larger odd multiple of π / 2 is more sensitive to intensity fluctuations than a smaller odd multiple of π / 2.

6. The method according to claim 4, wherein the fixed period t is set to drive a laser beam pulse of 9π / 2.

7. The method according to claim 1, wherein the step of adjusting the intensity of one or more of the laser beams includes controlling each of the laser beams using a radio frequency (RF) signal applied to each channel of a multi-channel acousto-optic modulator (AOM).

8. The method according to claim 7, wherein the step of controlling each of the laser beams using an RF signal includes adjusting the amplitude of the RF signal in response to identifying fluctuations in intensity in each of the respective laser beams.

9. The method according to claim 7, wherein the step of controlling each of the laser beams using an RF signal includes adjusting a first digital number A corresponding to the amplitude of the RF signal based on a second digital number B corresponding to the gain of a feedback loop of the plurality of separate measurements in response to identifying fluctuations in intensity in each of the respective laser beams.

10. The method according to claim 9, wherein the most significant bit (MSB) of the first digital number A is a bit representing the physical value of the amplitude of each of the RF signals.

11. Each laser beam has a diameter of about 1 μm to 1.5 μm at the position of each of the respective ions, the ions of the linear array of ions are separated from each other by a distance of about 5 μm, The method according to claim 1.

12. The method according to claim 1, wherein the method is performed at the end of an experimental cycle of the trapped ion system.

13. The method according to claim 1, wherein the method is performed during an experimental cycle of the trapped ion system.

14. The method according to claim 1, wherein the method is performed in less than 1 millisecond.

15. A system for stabilizing the intensity of a laser beam in a trapped ion system, a laser source configured to apply a linear array of laser beams to each of the ions of a linear array of ions in a trap, An imaging system configured to perform parallel measurements on each of the ions in response to the application of the laser beam, wherein the parallel measurements include a plurality of separate measurements for each ion in the linear array, each of the plurality of measurements for each ion indicates either the bright state or the dark state of the ion, and further, based on the plurality of separate measurements for each of the ions, configured to identify the fluctuations in the intensity of each of the laser beams at each ion. An optical controller configured to adjust the intensity of one or more of the laser beams in response to the fluctuations identified from the parallel measurements. A system comprising the above. **Claim 16** The system according to claim 15, wherein each of the plurality of separate measurements for a particular ion is a measurement of the response of that ion to each of the laser beams. **Claim 17** The system according to claim 15, wherein the optical controller is further configured to perform integral filtering of the plurality of separate measurements for each of the ions to identify the fluctuations in the intensity of each of the laser beams. **Claim 18** For each of the plurality of separate measurements for a particular ion, the optical controller is configured to prepare the ion in the quantum state │0>, and each of the laser beams is applied to that ion for a fixed period t that generates an equal superposition of the quantum states │0> and │1> when the intensity of the laser beam is correct and at a frequency that drives Rabi flopping of the ion between the quantum states │0> and │1>. The imaging system is configured to measure the response of the ion to the application of each of the laser beams as either "1" or "0", where "1" indicates the bright state of the ion and "0" indicates the dark state of the ion. The system according to claim 15. **Claim 19** The system according to claim 18, wherein the fixed period t is set to drive a laser beam pulse that is an odd multiple of π / 2, and a larger odd multiple of π / 2 is more sensitive to intensity fluctuations than a smaller odd multiple of π / 2. **Claim 20** The system according to claim 18, wherein the fixed period t is set to drive a laser beam pulse of 9π / 2. **Claim 21** The system according to claim 15, wherein the optical controller is further configured to control each of the laser beams using a radio frequency (RF) signal applied to each channel of a multi-channel acousto-optic modulator (AOM).

22. The system according to claim 21, wherein the optical controller is further configured to adjust the amplitude of the RF signal in response to identifying intensity fluctuations in each of the laser beams.

23. The system according to claim 21, wherein the optical controller is further configured to adjust a first digital number A corresponding to the amplitude of the RF signal with a second digital number B corresponding to the gain of the feedback loop of the plurality of separate measurements in response to identifying intensity fluctuations in each of the laser beams.

24. The system according to claim 23, wherein the most significant bit (MSB) of the first digital number A is a bit representing the physical value of the amplitude of each of the RF signals.

25. Each laser beam has a diameter of about 1 μm to 1.5 μm at the position of each of the ions, The ions of the linear array of ions are separated from each other by a distance of about 5 μm, the system according to claim 15.

26. The system according to claim 15, wherein the imaging system is configured to perform the parallel measurement at the end of the experimental cycle of the trapped ion system.

27. The system according to claim 15, wherein the imaging system is configured to perform the parallel measurement during the experimental cycle of the trapped ion system.

28. The system according to claim 15, wherein the imaging system is configured to perform the parallel measurement in less than 1 millisecond.

Images