text:
Scarce Autograph of General John Clem
Scarce Autograph of General John Clem - Also known as "Johnny Clem" The Drummer Boy of Shiloh. First Day Cover dated, "June 18th 1933 - Newark Ohio - Home of John L. Clem". Boldly signed on the front "Jno. L. Clem / U.S. Army" with notarized affidavit of how the Autograph was obtained!
John Lincoln Clem:
Residence was not listed; 11 years old.
Enlisted on 5/1/1863 at Nashville, TN as a Private.
On 5/1/1863 he mustered into "C" Co. MI 22nd Infantry
He was discharged on 9/19/1864
(Subsequent service in US Army)
He was listed as:
* POW 10/10/1863 Chattanooga, TN (Exchanged)
* Sergt
born 8/13/1851 in Newark, OH
died 5/13/1937 in San Antonio, TX
Buried: Arlington National Cemetery, Arlington, VA
Gravesite: 2-993
Colonel John L. Clem, who occupied Quarters #9 on Staff Post in 1902-03 and 1909-1911, was better known as "Johnny Shiloh" or as it says on his tombstone, "The Drummer Boy of Chickamauga."
A drummer boy in the Union Army at the age of 10, Clem earned a promotion to sergeant at the age of 12 by being the only soldier in his regiment to stand with his commander during a rout. By using Johnny as an example, the commander was able to re-form his unit and win the day. Clem's exploits were portrayed in the Disney film, "Johnny Shiloh."
John Lincoln Clem, was born in Newark, Ohio, on August 13, 1851. In May 1861 he attempted to enlist in the Third Ohio Volunteers but was rejected because of his youth. He attached himself unofficially to the Twenty-second Michigan Infantry and participated in the battle of Shiloh as a drummer. He enlisted shortly thereafter and participated in the battle of Chickamauga, in which, at the age of twelve, he shot a Confederate colonel who demanded his surrender. After the battle, the "Drummer Boy of Chickamauga" was promoted to sergeant and placed upon the roll of honor, the youngest soldier ever to be a noncommissioned officer in the United States Army.
In October 1863, Clem was captured in Georgia by Confederate cavalry while detailed as a train guard. The Confederate soldiers took his uniform away from him which reportedly upset him terribly--especially his cap which he said had three bullet holes in it. He was exchanged a short time later, but the Confederate newspapers used his age and celebrity status to show "what sore straits the Yankees are driven, when they have to send their babies out to fight us." After participating with the Army of the Cumberland in many other battles Gen. Thomas assigned Clem to his staff as a mounted orderly, until he was discharged in 1864.
He returned home and graduated from high school in 1870. After he attempted unsuccessfully to enter the United States Military Academy at West Point, President Ulysses S. Grant appointed him second lieutenant in the Twenty-fourth United States Infantry. Clem graduated from artillery school at Fort Monroe in 1875, transferred to the quartermaster department in 1882, and rose to the rank of major general by the time he retired in 1916. During his last years in the army he was the sole remaining Civil War veteran on active service.
Clem spent a number of his army years in Texas. From 1906 to 1911 he was chief quartermaster at Fort Sam Houston; after retirement he remained in Washington for a few years, then returned to San Antonio. He married Anita Rosetta French in 1875. She died in 1899, and he married Bessie Sullivan of San Antonio in 1903. Clem was the father of two children. He died in San Antonio on May 13, 1937, and was buried in Arlington National Cemetery.
CIVIL WAR HERO IS REPRIMANDED - Drummerboy of Chickamauga May Be Recalled for Blunder in Philippines
Washington. Sept 3. Colonel John M Clem, Chief Quartermaster of the Philippine Division, has been reprimanded by General Humphrey, Quartermaster General, for his action in turning into the United States Treasury the sum of $43,000 as unexpended balance of the appropriations on the last fiscal year, when, as stated at the war department, the money was actually needed to meet necessary expenses for army transportation and other work of the quartermaster's department.
It is said the service will be embarrassed by Colonel Clem's action, as it seriously interfered with the work of the quartermaster's department.
General Humphrey has informed Colonel Clem that his action is disapproved, and would not have been taken if he had had a proper conception of his duties. The money, having been turned into the treasury, is now beyond the control of the war department, and officials say it will have to be reappropriated by congress to meet existing obligations. It is not unlikely that Colonel Clem will be relieved of his present duties and recalled to the United States.
Colonel Clem is one of the heroes of the War of the Rebellion, and is known as the drummerboy of Chickamauga, because of his gallant conduct in the battle while serving as a drummer in the Twenty-Second Michigan Volunteer Infantry.
John Lincoln Clem (August 13, 1851 – May 13, 1937) was a United States Army general who served as a drummer boy in the Union Army in the American Civil War. He gained fame for his bravery on the battlefield, becoming the youngest noncommissioned officer in Army history. He retired from the U.S. Army in 1915, having attained the rank of brigadier general in the Quartermaster Corps; he was the last veteran of the American Civil War still on duty in the U.S. Armed Forces. By special act of Congress on August 29, 1916, he was promoted to major general one year after his retirement.
Born with the surname Klem in Newark, Ohio on August 13, 1851, he is said to have run away from home at age 10 in May 1861, after the death of his mother in a train accident, to become a Union Army drummer boy. First he attempted to enlist in the 3rd Ohio Infantry, but was rejected because of his age and small size. He then tried to join the 22nd Michigan, which also refused him. He tagged along anyway and the 22nd eventually adopted him as mascot and drummer boy. Officers chipped in to pay him the regular soldier's wage of $13 a month and allowed him to officially enlist two years later. Research has shown that Clem's claims about the 3rd Ohio and running away from home in 1861 (rather than in either 1862 or 1863) may be fictitious.
A popular legend suggests that Clem served as a drummer boy with the 22nd Michigan at the Battle of Shiloh. The legend suggests that he came very near to losing his life when a fragment from a shrapnel shell crashed through his drum, knocking him unconscious, and that subsequently his comrades who found and rescued him from the battlefield nicknamed Clem "Johnny Shiloh." The weight of historical evidence however suggests that Clem could not have taken part in the battle of Shiloh. The 22nd Michigan appears to be the first unit in which Clem served in any capacity, but this regiment had not yet been constituted at the time of the battle (mustering into service in August 1862 – four months after the Battle of Shiloh). The Johnny Shiloh legend appears instead to stem from a popular Civil War song, "The Drummer Boy of Shiloh" by William S. Hays which was written for Harpers Weekly of New York. The song was written following the Battle of Chickamauga in September 1863, and may have been written with Clem in mind because he had already become a nationally-known figure by that time.
Regardless of his entry into service, Clem served as a drummer boy for the 22nd Michigan at the Battle of Chickamauga. He is said to have ridden an artillery caisson to the front and wielded a musket trimmed to his size. In the course of a Union retreat, he shot a Confederate colonel who had demanded his surrender. After the battle, the "Drummer Boy of Chickamauga" was promoted to sergeant, the youngest soldier ever to be a noncommissioned officer in the United States Army. Secretary of the Treasury, later Chief Justice of the United States Supreme Court, and fellow Ohioan, Salmon P. Chase, decorated him for his heroics at Chickamauga. Clem's fame for the shooting is also open for debate, despite press reports supporting the story into the early 20th century. It is possible that he wounded Col. Calvin Walker, whose 3rd Tennessee opposed the 22nd Michigan towards the end of the battle.
In October 1863, Clem was captured in Georgia by Confederate cavalrymen while detailed as a train guard. The Confederates confiscated his U.S. uniform which reportedly upset him terribly—including his cap which had three bullet holes in it. He was included in a prisoner exchange a short time later, but the Confederate newspapers used his age and celebrity status for propaganda purposes, to show "what sore straits the Yankees are driven, when they have to send their babies out to fight us." After participating with the Army of the Cumberland in many other battles, serving as a mounted orderly, he was discharged in September 1864. Clem was wounded in combat twice during the war.
Clem graduated from high school in 1870. In 1871, he was elected commander/captain of the "Washington Rifles" a District of Columbia Army National Guard militia unit. After he attempted unsuccessfully to enter the United States Military Academy, after failing the entrance exam, President Ulysses S. Grant appointed him second lieutenant in the Twenty Fourth United States Infantry in December 1871. Clem was promoted to first lieutenant in 1874. Clem graduated from artillery school at Fort Monroe in 1875. He was promoted to captain in 1882 and transferred to the Quartermaster Department where he stayed for the rest of his career. He was promoted to major in 1895.
During the Spanish–American War in 1898 he served as depot quartermaster in Portland, Oregon as well as department quartermaster for the Department of Columbia. He then served in the occupation of Puerto Rico as depot and chief quartermaster in San Juan.
Clem was promoted to lieutenant colonel in 1901 and to colonel in 1903. He then served from 1906 to 1911 as chief quartermaster at Fort Sam Houston in Texas.
Clem reached the mandatory retirement age of 64 on August 13, 1915, when he was retired and promoted to the rank of brigadier general, as was customary for American Civil War veterans who retired at the rank of colonel, becoming the last veteran of the American Civil War to serve in the U.S. Army. On August 29, 1916, he was promoted on the retired list to the rank of major general.
After retirement he lived in Washington, D.C. before returning to San Antonio, Texas. He married Anita Rosetta French in 1875. After her death in 1899, he married Bessie Sullivan of San Antonio in 1903. Sullivan was the daughter of a Confederate veteran, leading Clem to claim that he was "the most united American" alive. Clem was the father of three children. Clem was a companion of the Military Order of the Loyal Legion of the United States and the Military Order of Foreign Wars.
He died in San Antonio on May 13, 1937, and was buried in Arlington National Cemetery in Arlington County, Virginia.
Through his military career Clem held the following ranks:
Musician and Lance Sergeant, Co. C, 22nd Michigan Infantry – 1 May 1863 to 19 September 1864
2nd Lieutenant – 18 December 1871
1st Lieutenant – 5 October 1874
Captain – 4 May 1882
Major – 16 May 1895
Lieutenant Colonel – 2 February 1901
Colonel – 15 August 1903
Brigadier General (Retired) – 13 August 1915
Major General (Retired) – 29 August 1916
MICHIGAN Twenty-Second Infantry. (Three Years)
On July 15, 1862, Governor Austin Blair issued General Order No. 154, directing the raising of six Regiments of Infantry, designated the Fifth Congressional District, composed of the counties of Livingston, Lapeer, Macomb St. Clair and Sanilac as the territory in which the Twenty-second Michigan Infantry Volunteers should be recruited, naming Pontiac as the place of rendezvous, and appointing ex-Governor Moses Wisner as Commandant of Camp. On August 8, 1862, the Field and Staff Officers were commissioned, and on the 31st of July following commissions were issued to the officers of the line. On August 29, 1862, the regiment having its full quota of officers and men, was mustered into the United States service. The officers were as follows:
Field and Staff.
Colonel, Moses Wisner, Pontiac.
Lieutenant Colonel, Heber Le Favour, Detroit.
Major, William Sanborn, Port Huron.
Surgeon, Abram P. McConnell, Pontiac.
Assistant Surgeon, Wells B. Fox, Hartland.
Adjutant, Edgar Weeks, Mt. Clemens.
Quartermaster, Thomas C. Boughton, Pontiac.
Chaplain, Asher E. Mather, Pontiac.
Line Officers.
Co. A. Captain, Ezra C. Hatton, Farmington. First Lieutenant, Edward M. Wisner, Pontiac. Second Lieutenant, William Albertson, Pontiac.
Co. B. Captain, Alonzo M. Keeler, Shelby. First Lieutenant, William Hulsart, Bruce. Second Lieutenant, Henry W. Howgate, Armada.
Co. C. Captain, John Atkinson, Port Huron. First Lieutenant, Jefferson J. Wilder, Capac. Second Lieutenant, John Sackett, Port Huron.
Co. D. Captain, Townsend C. Beardslee, Pontiac. First Lieutenant, Almeron S. Mathews, Pontiac. Second Lieutenant, Elijah Snell, Independence.
Co. E. Captain, Henry Carlton, Newport. First Lieutenant, Hazzard P. Wands, St. Clair. Second Lieutenant, Thomas C. Jackson, Ira.
Co. F. Captain, Alfred Ashley, New Baltimore. First Lieutenant, George W. Robertson, Mt. Clemens. Second Lieutenant, Heber B. Pearson, New Baltimore.
Co. G. Captain, Joseph Goetz, Mt. Clemens. First Lieutenant, William C. Stockton, Mt. Clemens. Second Lieutenant, Augustus Czizek, Mt. Clemens.
Co. H. Captain, Henry S. Dean, Green Oak. First Lieutenant, William A. Smith, Marion. Second Lieutenant, Louis Brown, Howell.
Co. I. Captain, Frederick W. Kimberk, Brighton. First Lieutenant, Walter Bowers, Lyon. Second Lieutenant, James Page, Lyon.
Co. K. Captain, Alexander G. Galbraith, Lexington. First Lieutenant, Henry Breidenbach, Lexington. Second Lieutenant, John A. Simons, Lexington.
September 4, 1862, the regiment left its camp on the Fair Grounds at Pontiac, and marched 1000 strong to the railroad depot, and took the cars for Detroit. Late that evening it took boat for Cleveland, at which place it arrived on the morning of September 5th. Here the regiment had its first issue of the rations on the march. From Cleveland it proceeded by rail to Cincinnati, where it arrived at midnight. It marched from the depot to the Fifth Street Market House, where at 2:00 o'clock a. m., September 6th, it ate breakfast provided by the city of Cincinnati. Before leaving the tables, the men were ordered to fill their haversacks, an order, however, it was never necessary to repeat during their term of service, when there were rations in sight. After breakfast, the regiment crossed the Ohio River on a pontoon bridge to Covington, Ky., and retired to rest on the paved streets of that town. Hard as was the bed, sweet was the sleep of these 1000 tired men, unaccustomed to carrying arms and knapsacks.
During the forenoon, the regiment was marched to the front, without ammunition, and formed in line of battle to resist a reported attack about to be made by the Confederate forces under the command of General E. Kirby Smith. Company H was ordered forward as skirmishers, to "feel the enemy," without a round of ammunition, which it proceeded to do until relieved by troops provided with ammunition; this line of battle was formed on a knoll in the center of a field of cabbage, and this first experience of the regiment in war was dubbed by the men as the "Battle of Cabbage Hill." After it was all over, the men laughed heartily, but while "the battle was on," it did not appear so funny. At midnight, the regiment retired in "good order," the battle having resulted in a great slaughter of Southern cabbage. Here the regiment was assigned to the Third Brigade, Second Division, Army of Central Kentucky, with which it served until April, 1863. The regiment remained in a fortification on Covington Heights until the evening of September 7th, when it marched one mile to the front and went into camp, where it remained until 1:00 o'clock a. m., when the bugle sounded "strike tents," and it marched back to Camp Wallace, when it had its first experience in building fortifications, a work in which it was engaged until September 18th, at which date it marched five miles to Florence, Ky., and camped on the Fair Grounds at that place. September 19th it marched nine miles south of Florence; on September 20th it moved eight miles in a southerly direction; September 21st it marched back over the route it had come to within twelve miles of Covington, Ky., and went into camp, naming its resting place "Camp Walton." From the 21st of September to October 9th it remained in this camp, forming line of battle from one to five times a night to meet threatened attacks by John Morgan's Cavalry. From Camp Walton, it moved to Williamston, naming its halting place "Camp Wells." There it remained until 11:00 o'clock p. m., October 14th, when all who were fit for duty proceeded to Cynthiana, where they arrived at 9:00 p. m., October 15th. The detachment left at Camp Wells, with the camp and garrison equipage, marched for Lexington, reaching that place October 21st. At Georgetown, through which town this detachment passed, it had an experience in returning slaves to their masters, which General Q. A. Gilmore ordered them to do.
With one exception he is the only United States officer that ever gave such an order to the Twenty-second Michigan Infantry. On the night of October 18th, Company H was ordered to proceed up the railroad and take possession of and hold Townsend's Bridge, which it did. That night, this Company was ordered "to proceed to Paris, take possession of the town, and hold same at all hazards." The company reached Paris at 4:00 o'clock a. m., October 18th, took possession of the town, and captured 100 of Humphrey Marshall's command. At 7:00 o'clock a.m., information was received from Lexington that John Morgan entered that place the evening before with 2,000 cavalry and a battery, and that he would move on Paris the next morning. Information of Morgan's plans was dispatched by handcar (three negroes furnishing the motive power), to Colonel Wisner at Cynthiana. At 2:00 p. m. of that day, Colonel Wisner left Cynthiana with the regiment and a battery, arriving in Paris at 7:00 p. m., making the march of eighteen miles in five hours; this was quick time for infantry, and a cavalry regiment dubbed the Twenty-second "The Marching Regiment." John Morgan, learning that Paris had been reinforced, did not attack the town. Upon a Confederate prisoner, a letter was found written by Humphrey Marshall to his wife, informing her that he had been ordered to move with his command out of Kentucky into Virginia by way of Pound Gap; this information was communicated to General Gordon Granger at Cincinnati, who ordered Colonel Wisner "to follow Humphrey Marshall, if he followed him to 'Davy Jones' locker." In obedience to this order, the regiment left Paris in pursuit of Humphrey Marshall. Fortunately, however, Humphrey Marshall's command of about 5,000 had so much the start of Colonel Wisner and his 800 men, that he did not overtake him. On this march the regiment passed through Lancaster to Athens, Ky., where it received orders to proceed to Lexington, where it arrived October 26, 1862. The detachment left at Camp Walton, with the camp and garrison equipage, had reached Lexington in advance of the main body.
When the regiment left Camp Walton on October 9th it moved "in light marching order," without tents, which every old soldier understands and never forgets, if, as in this instance, the march is late in the fall, and the ground most of the time covered with snow. Tents were a luxury the regiments had not enjoyed for seventeen days and nights.
From October 26, 1862, to February 21, 1863, the regiment remained in Camp Ella Bishop at Lexington, Ky. Here it learned the terrible fact that bullets are not the only danger incident to the life of a soldier. The four months spent in Lexington were months of suffering, with but little good to compensate therefor. The rigid discharge of picket duty in open fields, without shelter from the cold and storms of winter, without fires, sleeping on the damp ground, not permitted to use straw which was generously offered by citizens, brought sickness, suffering and death upon the regiment. Scores of noble men lie sleeping in the cemetery at Lexington, who died in consequence of exposure in picketing their own camps, to prevent Union soldiers from entering the town of Lexington, for there was not an armed enemy within the state at that time. Mrs. Sanborn, wife of Colonel Sanborn, spent the winter in camp, and the men of the regiment will never forget how much she did to alleviate their suffering--she was their good angel. On January 4th, 1863, Colonel Moses Wisner died after a lingering illness from disease contracted while in the faithful discharge of his duty; his last words were expressions of love for his country, sympathy and hope for the well being of his Regiment. Governor Blair in his annual message to the Legislature referred to the death of Colonel Wisner as follows:
"Intelligence has been received of another of the great sacrifices we make to save our country. My predecessor, ex-Governor Moses Wisner, Colonel commanding the Twenty-second Regiment of Infantry, died at his post of duty in Kentucky, on the 4th day of January. His conduct is his best eulogy. A man of great intellectual, as well as physical power, in the meridian of life, surrounded by all the comforts of family, home and friends, he obeyed the call of his country and took the field. Deeply imbued with a love of those free institutions which had done so much for his country and himself, he put away from everything but this service, and went forth at the head of his regiment to peril all in defense of the Union. As a commanding officer of patriotic volunteers, he was successful in an eminent degree, as he had been in all the walks of life. He died of the diseases of the camp, in the midst of his command, in the doing of his duty. More than this need not be said. For him the pomp and circumstance and the battle are no more. To his family and friends he leaves the rich remembrance of an honorable fame, and to the State he loved, the pride that she had so noble a citizen. To you, gentlemen, the representatives of the people, is committed the duty to fitly commemorate his services."
On the death of Colonel Wisner, Lieutenant Colonel Heber Le Favour was promoted to Colonel; Major William Sanborn to Lieutenant Colonel; and Captain Henry S. Dean to Major of the regiment.
On February 21st the regiment under the command of Colonel Le Favour marched from Lexington to Hickman Bridge on the Kentucky river, camping that night in the snow on what was known as the Scott farm. The next day it marched to Danville, where it remained until the afternoon of February 23rd, when it retraced its steps to Hickman Bridge, arriving there at midnight, tired and foot-sore, the men lying down on the bare ground without tents to sleep if they could. Scarcely had they rolled themselves in their blankets, when an order was received from General Gilmore, directing the regiment "to return to Lexington as soon as possible." At 1:00 o'clock a. m. it was moving in that direction. The camp equipage was unloaded from the wagons, and left under guard at the Bridge, and as far as possible, men who could march no longer were loaded into the wagons following the rear of the column. At daylight the regiment reached Nicholasville, where it took cars for Lexington. Upon arrival at the latter place Companies B, E, G, and K, under command of Major Dean, were ordered to proceed to Cynthiana to guard that place against a threatened attack. The detachment reached Cynthiana late in the afternoon of February 25th, tired and hungry, without rations, and no government stores to draw upon. Six wealthy Confederate sympathizers were requested to furnish and cook rations for these four companies.
To the credit of those parties be it said that during the two days they kept a boarding house for Union soldiers they "set a good table." February 26th this detachment was ordered to rejoin the regiment at Lexington where it remained until March 21st, when it was again ordered to proceed to Danville. A few days previous to this, Colonel Le Favour was thrown from his horse and very seriously injured; so seriously, that he was unable to resume duty until the May following. Under the command of Lieutenant Colonel Sanborn the regiment marched for Danville as part of a force, consisting of two regiments of infantry, one of mounted infantry, one of cavalry and a battery. A Confederate force under General Pegram had crossed the Cumberland River from Tennessee into Kentucky, with evident designs upon a depot of Government supplies stored at Danville. To the protection of these, the force of which the Twenty-second Michigan was a part, was hurrying. On March 21st the regiment marched from Lexington to Danville, distance forty miles, in eighteen hours; that night it was quartered in a church, the next morning it moved out one mile on the Stanford road and went into camp. In the meantime, the Government stores had been loaded into wagons, and the train started in the direction of Hickman Bridge, on the Kentucky river. At 11:00 o'clock a. m., March 25th, the enemy made his appearance, opening fire upon the regiment from his mountain howitzers; line was formed to meet the attack, but it was evident that he was after the train, and not a fight, as he immediately moved around the left of the line, taking a short cut through the fields to reach the pike on which the train was moving. The regiment was ordered to fall back through the town to protect the rear of the train. As it came up with the train, the enemy opened fire on the regiment from among the timber growing along side the pike. Lieutenant Colonel Sanborn moved the regiment by the flank behind a stone wall on the side of the pike, and commenced a vigorous return of the enemies' fire. Just then, General Carter and staff rode up, and ordered Lieut. Col. Sanborn to detach the three left companies of his regiment, under command of the Major, with a section of artillery to continue the fight and to move the rest of his command up with the train. During the afternoon, until the last wagon was safely across the Kentucky River, this detachment with the artillery moved to high ground, first on one side and then on the other of the pike, firing over the train to repulse the enemy in his repeated attacks upon the train. In this skirmish, the regiment had two men wounded and one captured. The regiment reached the Bridge late at night, tired, wet (it having rained all the afternoon), and hungry, and laid down in the mud and rain to sleep. The next morning, the regiment was ordered out on the Danville road to see if it could find the enemy, which it did, and posted a picket line; here it remained for two days, with occasional firing by the pickets. March 28th, the regiment returned to Nicholasville, and from there marched to Camp Dick Robinson and camped; the next day, it moved to Lancaster, and from there to Crab Orchard on March 30th. March 31st it moved toward Summerset, on the Cumberland River, in which direction Pegram was retreating with the plunder he had seized during the raid. That night, the regiment camped in the snow, without tents, at Buck Horn Creek. Whoever directed this campaign appeared to be full in the faith that infantry could keep pace with cavalry, and, in fact, the regiment was only eight miles behind Pegram's Cavalry, when it was overtaken by our mounted infantry, and 400 of his command and all his plunder captured at Summerset. April 1, 1863, the regiment was ordered to proceed to Nashville, Tenn., via Lebanon Junction. It arrived at the latter place April 9th, where it was to take cars for Nashville. It was detained at Lebanon thirty-two hours behind stacked arms, beside the cars upon which it was to proceed, because it refused to leave its colored servants in Kentucky. Brigadier General Manson, Post Commander, went so far as to order out a regiment of infantry, a squadron of cavalry, and a section of artillery to enforce his order. Colonel Sanborn firmly refused to obey the order, in which he had the support of every officer and man in the regiment. The matter was finally referred to General Burnside, at Cincinnati, who ordered the regiment "to proceed immediately to Nashville, and to stop quarreling about the nigger," which it did, taking its servants along. The regiment arrived in Nashville on the evening of April 13, 1863, and was assigned to the Third Brigade, Second Division, Reserve Corps, Army of the Cumberland. It remained at Nashville doing interior guard duty until September 5th. The Twenty-second Michigan and the Eighty-ninth Ohio were temporarily brigaded under the command of Colonel Heber Le Favour of the Twenty-second Michigan, and attached to General Whitaker's Brigade, General Steedman's Division, and General Gordon Granger's Reserve Corps. On Sept. 5, 1863, the Twenty-second Michigan left Nashville for Chattanooga, Tenn., with the following officers:
Lieutenant Colonel Wm. Sanborn, Commanding.
Captain Alonzo M. Keeler, Acting Major.
Abram P. McConnell, Surgeon.
Almeron S. Mathews, Adjutant.
Charles J. Bockins, Quartermaster.
Major Henry S. Dean was detailed on General R. S.
Granger's staff at Nashville, as Acting Inspector General.
The companies were officered as follows:
Co. A. First Lieutenant Wm. Albertson, commanding; Second Lieutenant Geo. W. Button.
Co. B. First Lieutenant Wm. Hulsart.
Co. C. Second Lieutenant Edgar G. Spalding, commanding; Captain John Atkinson was detailed on the staff of General R. S. Granger at Nashville.
Co. D. Captain Elijah Snell; First Lieutenant Lewis Drake; Second Lieutenant Wm. Willetts.
Co. E. Captain Hazard P. Wands; First Lieutenant Louis A. Allor.
Co. F. First Lieutenant George Robertson, commanding; Second Lieutenant Wm. B. Hamilton.
Co. G. Captain Joseph Goetz; First Lieutenant Charles Bassett; Second Lieutenant Edward E. Andrews.
Co. H. Captain Wm. A. Smith.
Co. I. Captain Fred W. Kimberk; First Lieutenant Hamilton J. Woodman; Second Lieutenant Lewis C. Mead.
Co. K. Captain Alex. G. Galbraith; First Lieutenant Henry Breidenbach.
The regiment arrived at Bridgeport, Ala., September 11th, and camped on Seven Mile Island until 7:00 a. m., September 13th. Here it left its baggage in charge of Lieutenants Henry Breidenbach and Wm. Hulsart and marched all day and night, passing over the base of Lookout Mountain at midnight. Here the men were so wearied with carrying their equipments, they slept on foot, jostling each other as they marched up and down the rocky slopes; then some one would waken, and break out in the words of the song, "John Brown's body lies mouldering in the grave, as we go marching on." At 11:00 a. m., September 14th, after a march of more than 40 miles, the regiment reached Rossville, Ga., where every man placed his gun in stack when the command "stack arms" was given. This fact coming to the knowledge of General Gordon Granger, he issued a special order, complimenting the regiment, and commending this example of good discipline to the rest of his command.
September 17th the Twenty-second Michigan, Eighty-ninth Ohio, and the Eighteenth Ohio Battery, with other troops, were ordered to make a reconnoissance in the direction of Ringold. As the regiment passed General Gordon Granger's headquarters, Company B, under command of Lieutenant Chas. Bassett, was detailed to guard the General's headquarters. Late in the afternoon, near Ringold, the enemy's pickets were driven in, and a few shells were thrown into the town, which received noresponse. Some hustling was observed in town, and considerable dust in the distance, as of troops marching away. So the force started back toward Rossville. About 11:00 p. m., after crossing Pea Vine Creek, pickets were posted and the regiment camped for the night, but just as it was being wrapped in comfortable sleep, shells came dropping into the camp. The pickets fell back upon the camp, a circumstance due to the orders given by a Staff officer, who posted them. The pickets
were sent back, the enemy fell back, and the regiment retired to rest, no harm being done. In the morning, the regiment renewed its march for Rossville, where it arived at 1:00 p. m., September 18th. On the afternoon of September 19th General Steedman ordered Colonel Le Favour to report with the Twenty-second Michigan and the Eighty-ninth Ohio to General Whitaker, who was then being attacked by the enemy near McAfee Church.
The Twenty-second Michigan took position on the left of the road and the Eighty-ninth Ohio on the right, with the Eighteenth Ohio Battery in the road in the rear, commanding a large field below and in front. This field gradually descended to a creek, bordered with thick brush and scattering timber. From this brush the enemy's sharpshooters came out into the open field to a log house, nearer to them than to the Union lines, with the intention of picking off some of the officers, or most exposed men. They soon opened fire through the spaces between the logs of the old house at the battery men. They seemed to think they had a safe place, until the Battery sent some shells crashing into their hiding place, setting the slivers flying about their heads. They were soon hugging the ground on the way back to the brush without having wounded a man. Night came on, and here the regiment slept on its arms for the first time. The night was cold, and without provision for it. No casualties resulting, except that one man accidentally discharged his rifle during the night, and disabled one of his toes. The morning was calm and beautiful, reminding one of the beautiful, sunny days of childhood, not a sound disconsonant with Holy Sabbath, which it was. Nothing in sight five miles away to betoken the fierce and bloody conflict of Chickamaugua that was that very day to test the courage of the Twenty-second Michigan Infantry. After roll call, and the men were getting ready to receive their rations, orders came from General Granger for General Steedman to report to General Thomas at once. Some men in their hurry to fall into line, took a handful of hardtack, others speared their bayonets into pieces of bacon, hoping to find time and opportunity to divide and share with each other for, at least, a slight breakfast.
Soon the artillery began to be heard, and the march was through fallen timber and among brush piles, so that the idea of breakfast was abandoned, the bacon dropped, and the ear was given to the roar of artillery. Soon an open field was reached--the road led past buildings that the day before had been used as field hospitals. Some dead were still on the ground beneath the trees. All that could be seen or heard was stimulating to the braver, and unnerving, perhaps, for the weaker soldier, but soon the enemy's fire across the line of march brought thoughts of present personal danger. By shelling, the enemy were attempting to prevent the passage of the column through between the heels of the horse shoe shape into which General Thomas' forces had been pressed by the enemy. The regiment had no time or opportunity to defend itself--it was hastening to the relief of others worse pressed.
A short halt in a corn field near General Thomas' position, that the officers might get directions to the place of direst necessity. While the men could not see over the rising ground, the roar of artillery and the rattle of musketry was too near and too fierce to make things attractive. Soon orders came to go in on General Brannan's right. Making its way over the low enclosure of the corn field into the woods, the regiment formed line of battle as it hurried up and to the right of Snodgrass Hill, and charged over Horse Shoe Ridge, led by Lieutenant Colonel Wm. Sanborn, down into the thickest of the fight. Captain F. W. Kimberk of Company I did not participate in this charge; Second Lieutenant Lewis C. Mead took command of the Company; First Lieutenant Hamilton J. Woodman being in command of a detail guarding a supply train. Here the Twenty-second Michigan, with the Eighty-ninth Ohio on its right, under Colonel Le Favour, met a most deadly fire from the enemy--the slaughter was fearful--Lieutenant Colonel Sanborn was seriously wounded while leading the regiment and taken to the rear; Captain Wm. A. Smith was mortally wounded, Color Sergeant Philo Durkee of Company A was struck in the breast by a grape shot-he fell mortally wounded--clasping the flag in his arms, he sealed his devotion to it with his blood upon its folds. Corporal Richard A. Stansell of Company H took the flag from the dying grasp of Sergeant Durkee, and gave his life for the flag, a musket ball passing through his brain. Corporal Pearl Mitchell of Company F raised the flag amidst the storm of shot and shell that soon carried away his left arm; Corporal Jonathan Vincent rushed for the flag and defiantly waved it in the face of the enemy--he fell severely wounded; Colonel Le Favour, coming up at this instant, shouted "Take up the flag." First Sergeant Wm. F. Atkinson took the flag and handed it to Sergeant Oscar Kendall, who, with a knowledge of the enemy's determination to put down the flag, threw away his rifle, took the flag, and with a courage and daring none can describe, planted it by his side, and stood as erect and fearless as God ever made man to stand for any cause. Corporal Fred Herger and one or two others of the Color Guard were killed; Captain A. M. Keeler, having taken command of the regiment after Lieutenant Colonel Sanborn was wounded, received orders to move the regiment back into line on the crest of the ridge where it joined Snodgrass Hill. There was a cessation of firing, and after closing up the broken ranks, the men were ordered to lie down. Within thirty minutes after the crest was occupied, two divisions of Longstreet's veterans furiously charged the Union lines. The men had been carefully instructed to select their men and fire at their hearts. The men lying upon the ground until the enemy were within easy range, quickly sprang to their feet and dealt them a fearful slaughter. For a minute the
firing was terrible, but the enemy were soon driven back. Captain Snell was mortally wounded and taken to the rear. Captain Galbraith was wounded but refused to be taken off the field. He asked Captain Keeler to examine the wound, who unbuttoned his waistbands, and putting his fingers in the wound found that the ball had been stopped by a bone and had glanced out. Captain Galbraith went back to his Company; Captain Goetz received a wound in his face, and brushing his hand over the wound went right in again; Lieutenant Spaulding was wounded, but remained with his Company. Word was sent back that the Twenty-second was nearly out of ammunition. Soon the enemy reformed, and came again determined to stay--the attack was desperate, and lasted for some minutes. The reply was brave and telling, and a charge broke and drove the enemy back again.
Although the day before the Twenty-second had taken the precaution to put forty rounds in each cartridge box and forty rounds in the pocket of each man, the long continued firing had entirely exhausted the ammunition of many men--more was called for. An Orderly came back, saying "Ammunition would be sent, but hold the ground at all hazards." What ammunition the men had was divided, and all that could be gathered from the cartridge boxes of the dead was distributed. General Granger, anticipating another charge at this point, sent orders, "to stand firm and use the cold steel." After an ominous silence of a few minutes, the enemy came again, but a well directed fire, followed by a charge, broke the last line of the day. Of this last attack, General Longstreet, on visiting the Chickamauga battlefield, in an article in the Atlanta Constitution, under the heading, "After Twenty Years," is made to say: There, pointing to Snodgrass Hill, "On that crest occurred the last collision of the day. A body of Federals made a gallant rally--it was then night--I threw my men on them, and my men melted away like shadows into darkness." Here ended one of the fiercest battles of the Civil War. The Twenty-second Michigan and Eighty-ninth Ohio were still holding the ground they had been ordered to hold at all hazards and with the cold steel. It was so dark no further fighting seemed possible, so dark that the blue could not be distinguished from the gray. Colonel Le Favour, coming near Captain Keeler asked, "have you any orders for the Twenty-second Regiment?" He replied, "I will go to the right and see." He did not return, but in a few minutes a body of troops were seen coming from the right, one from the left, and another from the rear. Then came an order to "throw down arms." The moon was rising, but there was no ammunition to give a ray of hope. This was the first intimation of defeat that the Twenty-second had received, and there was no alternative. The Sixth Florida, Fifty-fourth Virginia of Triggs Brigade, Preston's Division and Buckner's Corps, took the arms and colors of the regiment. As the Twenty-second was being taken away, another regiment of the enemy fired on them and their guards. The guard ordered "Lie down," and shouted, "You are firing on your own men." Nearly all dropped to the ground to escape a second volley. Lieutenant Albertson and John L. Clem, the regimental marker, feigning themselves dead, did not rise, and thus escaped a long imprisonment. The regiment was taken to General Preston's headquarters. He called for the ranking officer. Captain Keeler went forward, and was asked to what Regiment, Brigade, Division and Corps his command belonged, and what position it occupied in the fight? He asked how many men Gordon Granger had in his Corps. When told that there were between three and four thousand, he replied, "That's a damned lie; we know he had more men than that. Well, you fought like devils, we will use you well." Colonel Le Favour was captured with the Eighty-ninth Ohio when he went to the right for orders; the Twenty-second was taken after that, while waiting for orders. It claims that it was the last regiment to leave the line of battle, and that orders to leave the field should have been sent to this Brigade, while all the others were withdrawn, unless it was necessary to sacrifice this command to save the rest of the army. All the other Federal forces had received orders to retire to Rossville, and had gone. The last orders received by these two regiments were "to hold the ground at all hazards; stand firm and use the cold steel." The Twenty-second Michigan, when taken prisoners, numbered 178 men and 14 officers. That morning 500 men and officers answered to roll call. These prisoners were delivered at Libby Prison on the evening of September 30th, 1863. The next morning, the enlisted men were taken to other buildings, and afterwards sent to Belle Isle, Danville, Florence and Andersonville prisons, where nearly half of some of the companies died of starvation and cruel exposure. No pen can describe their inhuman treatment; the officers were kept in Libby Prison until May, 1864, and then moved to Macon, Ga., whence they were taken to Charleston, S. C., and placed under fire of the Union guns in the siege of Charleston; from there they were taken to Columbia, S. C., until Sherman's march compelled the enemy to move them to Charlotte and Raleigh, N. C., where they were paroled and passed through the Union lines near Wilmington, N. C., March 1, 1865. While every officer and man of the Twenty-second Michigan who participated in the battle of Chickamauga did his whole duty, Lieutenant Colonel Sanborn was conspicuous for his bravery. General Whitaker in transmitting a copy of his report of the battle to Governor Blair, speaks as follows of Colonel Le Favour and the conduct of the regiment:
Fort Whittaker, Opposite Lookout Mountain, Near Chattanooga, Oct. 12, 1863.
Sir: Having had the honor of commanding the Twenty-second Michigan, Colonel Le Favour, in my brigade, in the battle of Chickamauga, and being personally observant of their undaunted heroism, let me urge you for the good of the service, and as a reward to a chivalrous officer, to use your influence for the promotion of Colonel Le Favour.
Respectfully yours, etc.,
W. C. WHITAKER, Brigadier General Fourth Army Corps.
Inventory Number: DOC 119


text:
/*
  Write a program to read a value of days and
  convert it into year, month and day.
*/

#include <stdio.h>
int main() {
  int day, year, month, remDays;
  printf("Enter a number of days: ");
  scanf("%d", &day);

  // Calculate year
  year = day / 365;

  // Calculate remaining days
  remDays = day % 365;

  // Calculate month
  month = remDays / 30;

  // Calculate remaining days
  day = remDays % 30;

  printf("\n Year: %d, Month: %d, Day: %d", year,month,day);
  return 0;
}

redpajama_set_name: RedPajamaGithub


The U.S. uranium mining industry has been devastated. This year, we expect it will provide less than 2 percent of the uranium that our country's nuclear power plants need to produce 20 percent of our electricity — the lowest U.S. supply level since before the Cold War.
This is no accident. We believe this is a deliberate result of strategies by rival countries to increasingly dominate the global nuclear marketplace and undercut U.S. national and energy security.
More than one-third of uranium imports now come from state-sponsored enterprises in Russia and its satellites. That number is expected to increase as imports from allies such as Canada, Australia, and Namibia decrease. These government-owned industries employ what many would consider to be unfair trade practices that flood the global market with cheap uranium and nuclear fuel. Now China is following in their footsteps.
The Department of Defense-led analysis of the U.S. defense industrial base ordered by President Trump has been released. It describes how the Chinese government leverages its monopoly on certain natural resources to undermine the United States. China's goal, according to many experts, is to force U.S. suppliers in critical industries out of business.
The strategy is working. Six uranium mines in the U.S. have been forced to close in recent years because of artificially low prices. Allied uranium mining has been felled by the same geopolitical weapon. In Canada, only one uranium mine remains in operation, down from four in 2014. Soon, one of the largest uranium mines in Australia will shut down.


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text:
\section{Introduction}
Cold atoms in optical lattices\cite{re:Bloch2008,re:Lewenstein2012} currently
represent one of the best candidates for the quantum simulation
\cite{re:Buluta2009,re:Bloch2012,re:Georgescu2014} of bosonic and fermionic systems, given the
extreme level of tunability of the microscopic Hamiltonian
parameters, such as tunneling and interactions. Recent
experimental advances allow for site-resolved detection
 measurements\cite{re:Bakr2010,re:Sherson2010}, the simulation
of artificial gauge fields\cite{re:jaksch2003creation,re:lin2009synthetic,re:Dalibard2011,re:Struck2012},
and also the possibility to control
the behavior of the optical lattice under the influence of high
frequency electromagnetic fields\cite{re:Lignier2007,re:Zenesini2009}.
In addition, intriguing phenomena such as
Bloch oscillations and
the formation of repulsively bound pairs have been observed\cite{re:Winkler2006}.
A particularly interesting phenomenon is provided by the coherent
 destruction of tunneling, in which
driving the system with an oscillating field has the effect of
renormalizing the intersite tunneling amplitude. By tuning the amplitude
 of the driving field, it has been theoretically and experimentally
demonstrated that one can induce the superfluid-Mott insulator transition\cite{re:eckardt2005superfluid}.

Theoretically, it has been shown that new exotic interactions and
correlated hopping terms\cite{re:Diliberto2011} can
be effectively generated by the external electromagnetic fields.
These achievements have opened a new line of research, termed Floquet
engineering. Indeed, both
bosonic and fermionic Hubbard models are currently
theoretically and experimentally investigated, with the possibility of
exploring new exciting physics out of equilibrium.

In Refs.~[\onlinecite{re:Creffield2008,re:Creffield2011}], it has been demonstrated that a high-frequency monochomatic external field can be used not only
to control the hopping amplitude of cold atoms trapped in a lattice,
but also the phase of their wave function. In a one dimensional optical lattice of bosonic
atoms described by a Bosonic Hubbard model, the authors showed that
a coherent matter current can be generated. The current is induced
by a shift in the momentum distribution function's central peak, which is proportional to the amplitude of the driving force.

In this paper, we apply the same periodic perturbation to a
fermionic optical lattice described by the Fermi-Hubbard model.
We explore the particular regime of parameters where there
is an attractive interaction between the fermionic particles and
explore the possibility of realizing and
manipulating exotic pairing phases.
In this context, Fulde-Ferrell (FF) and Larkin-Ovchinnikov (LO) phases
recently attracted a great deal of interest from both the experimental\cite{re:Zwierlein2006fermionic,re:Partridge2006pairing,re:Liao2010spin}
and the theoretical community\cite{re:Orso2007,re:Feiguin2007,re:Casula2008,re:Luscher2008,re:Rizzi2008,re:Tezuka2008,re:Feiguin2012bcs,re:Guan2013}.
In the original idea of Fulde and Ferrell, Cooper pairs form
with finite center-of-mass momentum\cite{re:FuldeFerrel1964}. Larkin and Ovchinnikov
formulated a related proposal in which the superconducting order parameter
oscillates in space\cite{re:Larkin1964}.
These ideas are closely related, because the inohomogeneus order parameter\cite{re:Yang2001,re:Casalbuoni2004}
may be interpreted as an interference pattern between condensates
with opposite center-of-mass momenta. In our case, we find a pairing state similar to a FF state,
consisting of a condensate which moves as a coherent matter field
of Cooper-like pairs. In particular, we find that the center-of-mass momentum
of the condensate can be tuned by changing the amplitude and the frequency
of the external drive.

Using the Floquet
formalism\cite{re:Bukov2014,re:Bukov2014a,re:Eckardt2015,re:Bukov2015rev,re:Mikami2016},
we also derive an effective time-independent Hamiltonian for the system in the presence
of a high-frequency driving force, and compare the results coming from
this effective theory by solving the model numerically using the time
dependent Density Matrix Renormalization Group method (tDMRG) \cite{tdmrg1,tdmrg2}.

We finally investigate the pairing properties of the system
by using different quenching protocols. We propose to abruptly change
either the interaction between the fermions,
or the parameters of the external driving potential independently.
These protocols are investigated numerically using tDMRG and
could be experimentally accessible in current experimental setups.

The paper is organized as follows. In Section II we present the model
in different reference frames and apply Floquet theory in the rotating frame.
In Section III we show that the numerical results
obtained with tDMRG are in agreement with Floquet theory.
In Section IV we explore different quenching protocols for
the characterization of the exotic pairing phase found, and we finally summarize our results in the conclusions.

\section{MODEL}

In this work, we consider a one dimensional Fermionic Hubbard chain
subjected to an external time dependent perturbation.
The Hamiltonian in the laboratory reference frame is given by
$H_{lab}(t)=H+V(t)$, where $H$ is given by the
one dimensional Fermi-Hubbard model
\begin{equation}\label{HH}
H=-J \sum\limits_{\langle i,j\rangle, \sigma}(c^\dagger_{i,\sigma}
c_{j,\sigma} + h.c.) + U \sum\limits_{i} n_{i,\uparrow}
n_{i,\downarrow},
\end{equation}
while the tilting perturbation is given by

\begin{equation}
\label{Shak}
V(t) = \theta(t) A \sin(\omega t+\theta) \sum\limits_{i,\sigma}
i n_{i,\sigma},
\end{equation}
and constitutes a linear potential ramp applied to the sites of
the lattice suddenly turned on at time $t=0$. Notice that this
driving protocol is different from the one adopted in
Ref.~[\onlinecite{re:Creffield2008}], where the amplitude was linearly ramped on in time.
In this paper, we assume that the tilting perturbation
is sinusoidal, with phase $\theta=0$, amplitude $A$ and frequency $\omega$.
In this case, as was observed in Ref.~[\onlinecite{re:Creffield2008}] and as will be clear below,
the effects of the phase of the drive are maximum, giving a non
trivial contribution in the effective Floquet description.
In the Eq.~(\ref{HH}), $J$ is the hopping amplitude between
nearest neighbor sites
(indicated by $\langle i,j\rangle$), $U$ parametrizes the repulsion between atoms,
and $c^\dagger_{i,\sigma}$ ($c_{i,\sigma}$) is the standard
fermionic creation (annihilation) operator on site $i$ with spin
${\sigma}$ ($\bar{\sigma}$ indicates the opposite of $\sigma$),
while $n_{i,\sigma}=c^\dagger_{i,\sigma} c_{i,\sigma}$ is the fermionic
occupation operator. The Planck constant is set to
$\hbar=1$, the lattice parameter $a=1$, and all of the energies
are in the units of the hopping $J$. These definitions
automatically set the time unit to $\hbar/J$.
Throughout this paper we consider an attractive interaction between
the fermions $U<0$, a lattice with open boundary conditions and filling $N/L=2/3$, where $N$ is the
 total number of fermions and $L$ is the system size.
It is well known that the low energy physics of the model (\ref{HH})
can be described in terms of a
gapped spin mode and a gapless charge mode. The system belongs to the Luther-Emery
universality class and superconducting pair correlations are dominant,
\begin{equation}\label{Pair}
P_{i,j}=\langle \Delta_i^{\dag} \Delta_j\rangle \propto |i-j|^{-1/K_{\rho}}
\end{equation}
with an non-universal power law decay with exponent $K_{\rho}$
determined by the interaction $U$. The main purpose
of this paper is to investigate the properties of the pairing
correlation function in Eq.~(\ref{Pair}) in the presence of the
driving (\ref{Shak}).

\subsection{Laboratory, rotating reference frames, and Floquet approximation}
It is useful to remind how an observable quantity is calculated in
the laboratory reference frame. At time $t=0$, the ground state
$|\Psi_{0}\rangle$ of $H$ is initially calculated. Afterward,
the time evolved state $|\Psi_{lab}(t)\rangle= U_{lab}(t,0)
|\Psi_{0}\rangle$, where $U_{lab}(t,0)=\mathcal{T}\Big[\textrm{exp}(-i\int_{0}^{t}H_{lab}(t')dt')\Big]$,  is computed under the action of the entire
time dependent Hamiltonian $H_{lab}(t)$, $\mathcal{T}$ being the time-ordering operator. Assuming that the ground state
is normalized, an observable $O$ is
then calculated computing the expectation value
\begin{equation}\label{Obslab}
\langle O \rangle_{lab} = \langle \Psi_{lab}(t) |O| \Psi_{lab}(t)\rangle.
\end{equation}

We now consider the formulation of the problem in the ``rotating''
reference frame. We apply to the Hamitonian $H_{lab}(t)$ the
following time dependent canonical transformation
\begin{equation}
R(t)=e^{-i f(t) \sum_{j}j {\hat n}_{j}},
\end{equation}
where
\[
f(t)=A\int_{0}^{t}d\tau\, \text{sin}(\omega\tau)=A{1-\cos(\omega t)\over \omega}.
\]
In this way, the rotated Hamiltonian $H_{rot}(t)=R^{\dag}(t)H_{lab}(t)R(t)-iR^{\dag}(t)\partial_{t}R(t)$ is still time dependent and assumes the form
\begin{align}\label{Hrot}
H_{rot}(t)&=-J\sum\limits_{\langle i,j\rangle, \sigma}(e^{-i f(t)} c^\dagger_{i,\sigma}
c_{j,\sigma} + h.c.) \nonumber\\
&+ U \sum\limits_{i} n_{i,\uparrow}
n_{i,\downarrow},
\end{align}
where the hopping term acquires a complex nontrivial phase, which is the
central quantity in this paper. It is clear
 that, at $t=0$, $H_{lab}(0)=H_{rot}(0)$. Therefore, once the ground  state
 $|\Psi_{0}\rangle$ is computed, the time evolved state in the rotating frame,
 $|\Psi_{rot}(t)\rangle$, which can be obtained at each time $t$ from the laboratory
frame state vector
 with $|\Psi_{rot}(t)\rangle=R(t)|\Psi_{lab}(t)\rangle$ reads $|\Psi_{rot}(t)\rangle=
 U_{rot}(t,0)
|\Psi_{0}\rangle$, where $U_{rot}(t,0)=\mathcal{T}\Big[\textrm{exp}(-i\int_{0}^{t}H_{rot}(t')dt')\Big]$.
An observable can be obtained by evaluating the following expression
\begin{equation}\label{Obsrot}
\langle O \rangle_{rot} = \langle \Psi_{rot}(t) |R(t)OR^{\dag}(t)| \Psi_{rot}(t)\rangle,
\end{equation}
where the observable operator should be correctly
transformed to the rotated frame.
We now apply Floquet theory to the rotated Hamiltonian Eq.~(\ref{Hrot}).
According to Floquet theorem, the time evolution operator
$U(t_1,t_2)=\mathcal{T}\Big[\textrm{exp}(-i\int_{t_1}^{t_2}H(t)dt)\Big]$
of any time-periodic Hamiltonian $H(t)=H(t+T)$ can be written
in the form
\begin{equation}
U(t_1,t_2)=\textrm{exp}[-iK_{F}(t_2)]\textrm{exp}[-iH_{F}(t_2-t_1)]
\textrm{exp}[iK_{F}(t_1)],
\end{equation}
where $H_F$ is an effective time independent Hamiltonian
describing the long time dynamics of the system, called the Floquet
Hamiltonian, while $K_{F}(t)=K_{F}(t+T)$ is a time-periodic operator (called
kick operator) describing the fast evolution in one period of time $T=2\pi/\omega$.
It is well known that in the limit of very large frequency or small period
one can calculate the Floquet Hamiltonian and the kick operator perturbatively using a high frequency expansion. In this paper, we use a particular van Vleck expansion, which is invariant under the choice of the driving phase {\em in the rotating frame} ~\cite{re:Bukov2014,re:Eckardt2015,re:Mikami2016}:
\begin{eqnarray}
H_{F}^{(0)}&=&H_{0}=\frac{1}{T}\int_{0}^{T}H(t)dt\\
H_{F}^{(1)}&=&\frac{1}{\omega}\sum_{m=1}^{\infty}\frac{1}{m}[H_{m},H_{-m}]=\\
&=&\frac{1}{2!T i}\int_{0}^{T}dt_{1}\int_{0}^{t_1}dt_{2}f(t_1-t_2)[H(t_1),H(t_2)]\nonumber \\
K_{F}^{(0)}(t)&=&0\\
K_{F}^{(1)}(t)&=&\frac{1}{i\omega}\sum_{m\neq0} \frac{e^{i m \omega t}}{m} H_{m}
\nonumber \\ &=&-\frac{1}{2}\int_{t}^{t+T}dt'g(t-t')H(t'),
\end{eqnarray}
where we decomposed the Hamiltonian in a Fourier series
 $H(t)=\sum_{m=-\infty}^{\infty}H_{m}e^{i m \omega t}$, with $H_{m}$
operator-valued coefficients; the functions $f$ and $g$ are defined as $f(x)\equiv(1-2x/T)$
and $g(x)\equiv(1+2x/T)$.
Recently, the importance of the kick operators in the
stroboscopic and non-stroboscopic dynamics of Floquet systems has been highlighted
by several authors\cite{re:Bukov2014,re:Bukov2014a,re:Eckardt2015,re:Bukov2015rev}.
For the rotated Hamiltonian in Eq.~(\ref{Hrot}), the zero-order Floquet Hamiltonian
is given by
\begin{equation}\label{HF}
\begin{aligned}
H_{F}^{(0)}&=-J J_{0}(A/\omega)\sum\limits_{\langle i,j\rangle,
\sigma}\Big(e^{-i\frac{A}{\omega}}c^\dagger_{i,\sigma}
c_{j,\sigma} + h.c.\Big) \\
&+U \sum\limits_{i} n_{i,\uparrow}
n_{i,\downarrow},
\end{aligned}
\end{equation}
where the amplitude of the hopping matrix element is renormalized by the Bessel function
$J_{0}(A/\omega)$. Notice that the hopping in the Floquet
Hamiltonian is complex acquiring a phase $\phi=-A/\omega$.
This phase comes from transformation to the rotating frame and would be
absent for the cosinusoidal driving field~\cite{re:Creffield2008},
which is equivalent to the shift in the rotating frame. The above equation
has been derived using the Jacobi-Anger expansion
\begin{equation}
\textrm{exp}\Bigg[i \frac{A}{\omega} \cos(\omega t)\Bigg] = \sum_{m=-\infty}^{\infty}
i^{m}J_{m}(A/\omega) e^{i m \omega t},
\end{equation}
which allows one to simply calculate the $m$-th component of the Fourier series
operator-valued coefficients of the Hamiltonian
\begin{align}\label{eq:FourierDec}
H_{m}&=-J J_{m}(A/\omega) \sum\limits_{\langle i,j\rangle,
\sigma} \Big(e^{-i\frac{A}{\omega}}i^{m} c^\dagger_{i,\sigma}c_{j,\sigma}
+ h.c.\Big)\nonumber\\&+\delta_{m,0}U \sum\limits_{i} n_{i,\uparrow}n_{i,\downarrow},
\end{align}
where it is important to point out that the interaction part contributes only for $m=0$.
Since at zero order of the high frequency expansion the kick operator is zero,
the evolution operator in the rotating frame $U_{rot}(t,0)$ becomes simply
\begin{equation}\label{eq:Urot0}
U_{rot}^{(0)}(t,0)=\textrm{exp}(-i H_{F}^{(0)}t).
\end{equation}
Using the Fourier decomposition in Eq.~(\ref{eq:FourierDec}), it is easy to prove that
at first order one has
\begin{align}
H_{F}^{(1)}&=0,\nonumber\\
K_{F}^{(1)}(t)&=\Big(i {J\over \omega} e^{-i A/\omega}\sum_{m\neq 0} e^{i m \omega t} \times\nonumber\\
&i^m\frac{J_{m}(A/\omega)}{m}\sum\limits_{\langle i,j\rangle,
\sigma}c^\dagger_{i,\sigma}c_{j,\sigma}\Big) +h.c. \label{eq:kick}
\end{align}
We see that the leading correction to the kick operator is proportional to $J/\omega$.
The time evolution operator $U_{rot}(t,0)$ at the first order of the high
frequency expansion reads
\begin{equation}\label{eq:Urot1}
U_{rot}^{(1)}(t,0)=\textrm{exp}[-iK^{(1)}_{F}(t)]\textrm{exp}(-i H_{F}^{(0)}t)
\textrm{exp}[iK^{(1)}_{F}(0)],
\end{equation}
with a nontrivial contribution coming from the kick operator. In particular, this contribution is nonzero at any time $t$ implying that there is always a non-zero kick to the initial wave function. Using the Floquet approximation, an observable can be obtained by evaluating the following expression
\begin{equation}\label{ObsFlo}
\langle O \rangle_{F} = \langle \Psi_{F}(t) |R(t)OR^{\dag}(t)| \Psi_{F}(t)\rangle
\end{equation}
where $|\Psi_{F}(t)\rangle=U_{rot}^{(k)}(t,0)|\Psi_{0}\rangle$, and $U_{rot}^{(k)}(t,0)$
should be calculated at the $k$th order of the high frequency expansion.
Notice that, as for Eq.~(\ref{Obsrot}) the observable operator should be correctly
transformed to the rotated frame.

\section{Numerical results}

We start considering an attractive interaction with strength $U=-4$, a large driving
frequency $\omega=20$, and an amplitude $A/\omega=0.5$.
Fig.\ref{fig1} shows the Fourier transform of the pair correlation function
defined in Eq.~(\ref{Pair}) as a function of time
\begin{equation}
P(k,t)=\frac{1}{L}\sum_{i,j}e^{ik(i-j)} \langle\Delta_i^{\dag} \Delta_j\rangle(t).
\end{equation}
Panel (a) is calculated in the laboratory reference frame with
Eq.~(\ref{Obslab}); in panel (b), Eq.~(\ref{Obsrot}) is used, while
in panel (c) we use the zero-order Floquet approximation as described in the
previous section, see Eq.~(\ref{ObsFlo}) where Eq.~(\ref{eq:Urot0}) has been used.
Notice that, at first order of the Floquet expansion, the only
extra contribution to the evolution operator comes from the kick operator
$K_{F}^{(1)}(t)$.
In the numerical simulations, the ground state $|\Psi_0\rangle$ of $H_{lab}(t)$ at $t=0$
is calculated using ground state DMRG.
We then use a third order Suzuki-Trotter decomposition of
the time evolution operator,\cite{re:Feiguin2004}
with a typical time step ten times smaller than
the period of the perturbation $\tau=0.1(2\pi/\omega)$.
In this work we put a great deal of emphasis in achieving a high degree of accuracy,
and for this purpose we limit ourselves to relatively small systems.
We consider chains up to $L=24$ sites long, keeping up to $m=1000$
states, enough to keep the total truncation error
(given by the accumulation of the DMRG truncation error and Trotter decomposition)
below $10^{-8}$.
We have also verified that the behaviour of the pair correlation function
does not depend qualitatively on the system size.
Therefore we can assume that our observations will be correct also in
thermodynamic limit.

Fig.~\ref{fig1} shows how,
for sufficiently small time $t\lesssim 3T$, the momentum of the Fermionic
condensate, originally at $k=0$, oscillates harmonically around a \emph{finite}
value $k^{*}$.
One can fit the motion of the central peak of the condensate in the Brilloin zone
with the relation $k^{*}(t)=2\frac{A}{\omega}(1-\cos(\omega t))$.
Panel (a) of fig.~\ref{fig1a} describes the magnitude of the condensate, $P_{\rm max}(t)=\max_k P(k,t)$, as a function of time
in the four schemes presented in the previous section.
As it should be the case, the laboratory and rotating frame results
coincide exactly. We also found that the behavior of the condensate peak can be fitted by a relation
$P_{max}=A+B\cos(Ct+D)$ in the Floquet first order approximation,
with $C\simeq 1.23|U|$.
\begin{figure}[h]
\centering \includegraphics[width=8cm]{Fig1.pdf}
\caption{(Color online) Pair correlations functions as a function of
momentum and time in the laboratory, reference frames, and zero-order Floquet
approximation. In this plot, a $L=12$ sites chain has been simulated using $m=400$
DMRG states, $U=-4$ and $A/\omega=0.5$.} \label{fig1}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[width=8cm]{Fig2.pdf}
\caption{(Color online) Panel (a): magnitude of the condensate shown in
the laboratory, rotating and Floquet first order approximation as a function
of time. The solid (dark yellow) line indicates the fit
$P_{max}=A+B\cos(Ct+D)$, as discussed in the main text.
Inset: local density at the center of chain of length $L=12$ as a function
of time (same filling and other parameters' value as in fig.~\ref{fig1}). Solid thick (black) line indicates data in the laboratory frame,
the solid thin (magenta) line the data in the zero-order Floquet approximation,
and the dashed (blue) line the data in the Floquet first order approximation.
Panel (b): Long time behavior of the
magnitude of the condensate. Notice the agreement between the laboratory frame and the Floquet approximation at first order.
Inset: Magnitude of the condensate as a function of time for a system of length $L=24$,
keeping up to $m=1000$ DMRG states with a maximum truncation error of $10^{-8}$.} \label{fig1a}
\end{figure}
The inset of panel (a) shows the local density at the center of chain of length $L=12$ as a function of time.
The data in the laboratory frame
(solid thick black line), which can be considered as an \emph{exact} result,
is compared with the zero and first order Floquet approximation
(solid thin magenta and dashed blue lines, respectively).
Even in the large frequency regime $J/\omega =0.05 \ll 1$,
one can see that the introduction of the kick operator
at first order in the Floquet expansion is crucial to reproduce
the out of equilibrium dynamics of the system
within a single period of the external drive.
Moreover, the results obtained in the laboratory frame
and the first order Floquet approximation agree
for all the times and system sizes investigated in this work.
Indeed, panel (b) of fig.~\ref{fig1a} studies the long time behaviour of the magnitude
of the fermionic condensate for a larger system size chain showing a robust
agreement between the two schemes.
In both panels of fig.~\ref{fig1a}, we have retained up to $|m|=5$ harmonics in the expression of the kick operator in eq.~(\ref{eq:kick}).

Fig.~\ref{fig2} shows the pair correlation function for different values of
the ratio $A/\omega$. One can notice that, by increasing the ratio $A/\omega$,
the \emph{average} position of the condensate peak moves towards the
edge of the Brilloin zone. Eventually, it gets reflected at the boundary zone edge
as displayed in Fig.~\ref{fig2a}b, where the average positions of the center
of the condensate for small times $t\lesssim5T$ are shown. By changing the
driving parameters in the interval $0<A/\omega\lesssim 3$ one can tune the central
peak of the Fermionic condensate in the entire Brilloin zone (see Fig.~\ref{fig2a}(b)).
This effect demonstrates the possibility of tuning the average momentum
of the condensate by changing the parameters of
the external driving.

\begin{figure}[h]
\centering \includegraphics[width=8.5cm]{Fig3.pdf}
\caption{(Color online) Pair correlations functions as function of momentum and time
calculated in the laboratory frame for different values of the ratio $A/\omega$.
In this plot, a $L=12$ sites chain has been simulated using $m=400$
DMRG states, $U=-4$.} \label{fig2}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[width=8.25cm]{Fig4.pdf}
\caption{(Color online) Panels (a-c): Magnitude of the condensate shown in
Fig.~\ref{fig2} as a function of time for different values of the ratio $A/\omega$.
Panel (b):
average position of the center-of-mass momentum of
the condensate as a function of $A/\omega$ extracted from Fig.~\ref{fig2}.
The solid (red) line is a guide to the eye. Panel (d):
Solid (red) line represent the real part of the hopping in the Floquet
Hamiltonian in Eq.~(\ref{HF}). Dashed (green) line is the imaginary part,
short dashed (blue) line indicates its modulus. The line $y=0$ is
drawn for convenience.} \label{fig2a}
\end{figure}
Panel (a) of Fig.~\ref{fig2} reproduces the trivial result for no driving as a reference.
Focusing with more attention on the four panels (b-e), we find (not noticeable by eye) that
the time-averaged position of the condensate in k-space at each period is \emph{not}
constant in time.
The average momentum of the condensate is slowly reducing its amplitude
as a function of the time. Correspondingly, we have studied the magnitude of the
center-of-mass of the condensate as a function of time, which
encodes the coherence of the Fermionic condensate. In Fig.~\ref{fig2a}(a) one can notice
that, after an initial transient for times $t\lesssim5T$,
the coherence of the condensate decreases monotonically. This effect reaches
its maximum for $A/\omega\simeq0.75$. For larger driving amplitudes, the system builds up
oscillations that survive at longer times, see the blue curve in Fig.~\ref{fig2a}(a).
 Interestingly, for a sufficiently large
ratio of $A/\omega$, the oscillations of the magnitude of the condensate at large
times are not suppressed anymore but are instead persistent, as shown in Fig.~\ref{fig2a}(c). In order to
understand this behaviour, it is important to consider the effective description
provided by the Floquet Hamiltonian.
Fig.~\ref{fig2a}(d) shows real, imaginary and modulus of
effective hopping parameter, $J_{eff}=J J_{0}(z)e^{-i A/\omega}$, in the Floquet Hamiltonian
in Eq.~(\ref{HF}). We notice that the modulus of
the effective hopping represents the well known renormalization of
the electronic band structure due to the periodic driving, which can result
in the dynamical fermionic localization at the zeros of the Bessel function $J_{0}(z)$.
In the interval $1\lesssim A/\omega\lesssim2.5$ the  modulus of the
effective hopping is significantly smaller than the other  energy
scale appearing in the Floquet Hamiltonian,
the Coulomb repulsion $U$, $|J_{eff}| \ll U$.
Interestingly, in this regime, the suppression of oscillations in the
magnitude of the  condensate are reduced, as seen in Fig.~\ref{fig2a}(c).

Fig.~\ref{fig2b} presents the Fourier transform of the data
shown in Fig.~\ref{fig2a}(a) and (c). Notice that the curves
obtained for $A/\omega\leq 1$ in Fig.~\ref{fig2a} are not time-periodic
so the corresponding Fourier transforms shown in Fig.~\ref{fig2b} present
a low frequency peak, which simply reflects this slow decay. For larger driving amplitude $A/\omega=1.0$ one can notice emergence of
two additional peaks at $\omega\pm U$ in the Fourier spectrum.
The remaining curves of Fig.~\ref{fig2b} are the Fourier transforms of the
data shown in Fig.~\ref{fig2a}(c). Same as before, the main feature of the curves
is the presence of the two peaks at $\omega\pm U$. Notice the appearance of the
spectral peak at $\omega'\simeq|U|$ together with a small contribution of two peaks at
larger frequency corresponding to higher harmonics, $2\omega\pm U$. The contribution
of high frequency harmonics is particularly significant at $A/\omega=2.4048$,
where the absolute value of the effective hopping is zero, see Fig.~\ref{fig2a}(d).
\begin{figure}[h]
\centering
\hspace*{-0.5cm}
\includegraphics[width=8cm]{Fig5.pdf}
\caption{(Color online)
Fourier transform of the curves shown in panel (a-c) of Fig.~\ref{fig2a}.} \label{fig2b}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[width=8cm]{Fig6.pdf}
\caption{(Color online)
Magnitude of the condensate for $A/\omega=2.4048$ and different values
of the Coulomb repulsion $U$, as indicated. The red curves represent fits of the
data according to the relation $P_{max}=A+B\sin(U t)\sin(\omega t)$.} \label{fig3}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[width=8cm]{Fig7.pdf}
\caption{(Color online)
Stroboscopic evolution of the magnitude of the condensate shown in Fig.~\ref{fig3}.
The red curves represent fits of the
data according to the relation $P_{max}=A-B\sin(U t+\phi)$.} \label{fig3b}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[width=8cm]{Fig8.pdf}
\caption{(Color online)
Fourier transform of the curves shown in Fig.~\ref{fig3}.} \label{fig3a}
\end{figure}
The results obtained can be interpreted as in Ref.~[\onlinecite{re:Bukov2016}]. In fact,
in the regime $U,\omega \gg J$, one can see that in a rotating frame defined by the
canonical transformation $V(t)=e^{-i (f(t) \sum_{j}j {\hat n}_{j}+Ut\sum_{j}
n_{\uparrow,j}n_{\downarrow,j})}$, the electron-electron interation
$U$ induces non-trivial phase shifts in the hopping amplitude.
Fig.~\ref{fig3} shows the magnitude of the condensate as a function of time
for $A/\omega=2.4048$ for different values of the
Coulomb repulsion $U$. Notice that
an accurate fit of the data can be obtained using the relation
$P_{max}=A+B\sin(U t)\sin(\omega t)$, showing \emph{beats} with frequencies
$\omega\pm U$.
The stroboscopic evolution at times equal to multiples of the period
$T=2\pi/\omega$ of the driving perturbation is shown in Fig.~\ref{fig3b}.
One can clearly observe that in a non-stroboscopic analysis, both the fast
(the frequency of the driving force $\omega=20$) and the slow (the Coulomb repulsion $U$)
time scales of the dynamics are resolved.
A Fourier analysis of the data provides more details on the non-stroboscopic evolution,
see Fig.~\ref{fig3a}. We find a clear contribution of at least
three harmonics, with pairs of peaks at $n\omega\pm U$ (with $n=1,2,3$).
For each pair, the distance between the two peaks is proportional to $2U$.
The intensity of the Fourier peaks decreases almost exponentially
with the order of the harmonics.
Indeed, the zero-order Floquet expansion
presented in the previous section for $A/\omega=2.4048$
corresponds to a Floquet Hamiltonian given just by the interaction term
$H_{F}^{(0)}\simeq U \sum\limits_{i} n_{i,\sigma}n_{i,\bar{\sigma}}$.
Therefore, this confirms that
the zero-order Floquet approximation governs the stroboscopic long
time evolution of the system.

In the next section, we will study the properties of the pairing states of the
system in the laboratory reference frame by using different
quench protocols.

\section{Characterization of the pairing phase: quenching protocols}

In this section we investigate the properties of the pairing phase by abruptly changing
either the interactions, or the parameters of the periodic driving force.
We first consider the quench protocol where we abruptly
 change the ratio $A/\omega$ of the driving
 but leaving the bare hopping and the $U$
attraction strength fixed. In practice, we change the values of the amplitude $A$ and keep
the frequency of the driving fixed. In Fig.~\ref{fig4}, we assume that before the quench
$A_1/\omega=0.5$ while after the
quench $A_2/\omega=2.4048$, which is the ratio corresponding to coherent destruction of
tunnelling. In panels (a) and (b), the pair correlations functions are shown as a
function of the momentum $k$ and the time $t$. In the panels (c) and (d) the magnitudes
of the peaks at zero momentum are shown for $t=nT$, with $n$ non-negative integer.
In panels (a) and (c), as also indicated by a vertical solid line, the quench is
performed at $t=5T$, that is after exactly $5$ periods of the driving force.
In panel (b) and (d), the quench is performed after $3.5$ periods of the driving.
Panels (a) and (c) show a very interesting behaviour. If, in a real experiment, one is
able to quench the amplitude of the driving at an instant of time that is an exact multiple of the driving
frequency $\omega$, one would measure, in average, just a shift in the
central peak of the pair distribution function, with no destruction of the condensate.
\begin{figure}
\centering \includegraphics[width=8.6cm]{Fig9.pdf}
\caption{(Color online) Panels (a)-(b): Pair correlation functions before and after
a quench of the amplitude of the driving from $A_1/\omega=0.5$ to $A_2/\omega=2.4048$.
In panel (a) the quench is performed at $t=5T$, while in panel (b) at $t=3.5T$.
Panels (c)-(d): Intensity of the peak at zero momentum of the same functions shown
in panels (a) and (b) evaluated at each period of time $T$. In this plot, a $L=12$ sites chain
has been simulated using $m=400$ DMRG states.} \label{fig4}
\end{figure}
On the contrary, if a quench of the amplitude of the driving is done in any instant of
time between $nT$ and $(n+1)T$ (we picked $3.5T$ as quenching time just for
convenience), the oscillation of the peak of the condensate picks up a \emph{random}
phase factor which determines a significant (about 80\%) suppression of the hight of the
condensate peak (see panel (d)).

We now consider the possibility of quenching the interaction between the fermionic
atoms loaded on the optical lattice. In panel (a) of Fig.~\ref{fig5},
we assume that the fermionic condensate is initially prepared with $U_1=-8.0$ and with driving
 parameters given by the ratio $A_1/\omega=0.5$, such that the center of the condensate
  is around $k^{*}=1$ in average.
At $t=5T$, we quench the interaction from $U=-8$ to zero, keeping
the parameters of the
driving fixed. After about a period of oscillation, the fermionic condensate is
almost completely suppressed, because of the lost attraction between the fermions.

\begin{figure}
\centering \includegraphics[width=8.5cm]{Fig10.pdf}
\caption{(Color online) Pair correlations functions as function of momentum and time
for different values of the ratio $A_2/\omega$. A quench in the interaction from $U_{1}=-8$ to $U_{2}=0$ is performed at $t=5T$. For $t<5T$, $A_1/\omega=0.5$.
In this plot, a $L=12$ sites chain has been simulated using $m=400$ DMRG states.} \label{fig5}
\end{figure}

\begin{figure}
\centering \includegraphics[width=8cm]{Fig11.pdf}
\caption{(Color online) Zero momentum contribution of the Pair correlations
 functions shown in Fig.~\ref{fig4} evaluated at each period of time $nT$,
with $n=0,1,2,..,14$. In this plot, $A_1/\omega=0.5$ and $U_{1}=-8.0$.} \label{fig6}
\end{figure}

\begin{figure}
\centering \includegraphics[width=8.5cm]{Fig12.pdf}
\caption{(Color online) Pair correlations functions before and after a quench at
$t=3.5T$ of the interaction $U$. Initially, $U_1=-4.0$,
while driving parameters are given by the ratio $A/\omega=0.5$.
In this plot, a $L=12$ sites chain has been simulated using $m=400$ DMRG states.} \label{fig7}
\end{figure}

A very non-trivial effect is visible if one increases the ratio $A_2/\omega$ after
the quench: the coherence of the condensate displays revivals
even if the bare interaction between the fermions is quenched to zero.
A stroboscopic measure of the coherence of the condensate is provided in the
panel (a) of Fig.~\ref{fig6}. The revival of the coherence of the condensate is
maximum around $A_2/\omega\simeq\pi/2$ and $A_2/\omega\simeq2.4048$. All the other cases
plotted, $A_2/\omega=1$, $A_2/\omega=2$, and $A_2/\omega=2.75$ show a decrease of
the magnitude of the condensate peak.
A qualitative explanation of the effects goes as follows: at the
time of the quench, the effective Floquet Hamiltonian of the system changes
accordingly with the new parameters of the driving. As demonstrated in Fig.~\ref{fig2a}(d),
the real part of the effective hopping vanishes exactly at $A/\omega\simeq\pi/2$
and $A/\omega\simeq2.4048$. As shown in Ref.~[\onlinecite{re:Creffield2011}] the real part of the effective
hopping is proportional to the spreading of a condensate prepared in real space,
which is then let evolved in time under the action of the driving.

We finally address a peculiar quenching protocol where we keep the
parameters of the driving fixed and, after preparing the condensate with a finite
average center-of-mass momentum, we quench the attraction strength
at time $t=3.5T$ (as indicated by the black vertical line).
Results are shown in Fig.~\ref{fig7}.
If the strength of the attraction between fermions is quenched to a value which is
larger in modulus than the initial value $U=-4$ (see panels for $U_2=-12$ and $U_2=-6.28$), the
center-of-mass momentum does not change, while the magnitude of the condensate peak
is increased (not shown in the plot). If the modulus of the strength of the attraction between the
fermions is decreased, not only the
coherence of the condense decreases, going to zero if $U_2=0$, but the average
center-of-mass momentum also drifts toward zero (see panels for $U_2=-2$
and $U_2=-1$).

\section{CONCLUSIONS}
In this work, we have studied the one-dimensional attractive Fermionic
Hubbard model subjected to a sinusoidal periodic drive.
We have assumed that the frequency of the driving force is large enough such
that the system is in the off-resonant regime, and Floquet theory can be applied.
To benchmark our numerical simulations, we have verified that a Van Vleck high-frequency expansion
in the rotating frame truncated to first order is able
to describe very well the numerical results obtained with time dependent
DMRG. In particular, at zero order,
the stroboscopic dynamics in a rotated reference frame is described by
an effective Floquet Hamiltonian with a complex phase in the hopping term
which introduces a shift of the condensate center-of-mass momentum.
We have also derived the \emph{micromotion} or \emph{kick}
operators and verified that their inclusion at first order of the
high-frequency expansion is crucial to understand
the dynamics of the system within one period of the external drive.

By tuning the amplitude and the
frequency of the driving force, one can realize an unconventional
Fermionic condensate with a finite center-of-mass momentum.
This pairing state is similar to a FF state,
because it represents a coherent matter field of Cooper-like pairs
at finite momentum.
In particular, the center-of-mass of the condensate can be tuned at
will across the Brilloin zone.
Moreover, we have shown that the coherence of the condensate, which is
encoded in the magnitude of the pair correlation function in momentum
space, can be enhanced by tuning the amplitude and
frequency of the drive to ratios close to those where one
obtains fermionic dynamical localization.

From the analysis of the real time evolution of the magnitude of the
condensate, we found that the stroboscopic dynamics is governed by
oscillations with period proportional to the electron-electron
attraction strength $U$, while the full time evolution presents
also oscillations given by the driving frequency $\omega$, creating
\emph{beats} at $\omega\pm U$.

Finally, we show that using different
quenching protocols, the condensate can be \emph{frozen}
at a particular finite center-of-mass momentum and coherence.
Moreover, the coherence of the condensate displays revivals
even if the bare interaction between the fermions is quenched to zero.
These protocols are within experimental reach.

Our work could pave the way for engineering out-of equilibrium FFLO-like phases
in multicomponent Fermi gases without population imbalance.

\section{ACKNOWLEDGMENTS}
This work was conducted at the Center for Nanophase
Materials Sciences, sponsored by the Scientific User Facilities
Division (SUFD), Basic Energy Sciences (BES), U.S. Department of Energy (DOE),
under contract with UT-Battelle. A.N. acknowledges support by the Center for
Nanophase Materials Sciences, and by the
Early Career Research program, SUFD, BES, DOE.
A.E.F. acknowledges the DOE, Office of Basic Energy
Sciences, for support under grant DE-SC0014407. A.P. was supported by
NSF DMR-1506340, ARO W911NF1410540 and AFOSR FA9550-16-1-0334.

redpajama_set_name: RedPajamaArXiv


text:
Home > North Korea remains on top of watch list for Christian persecution
Published January 8th, 2010 by Peter Elliott
North Korea remains on top of watch list for Christian persecution
For the eighth consecutive year North Korea tops the list of 50 countries regarded as the worst persecutors of Christians. Open Doors USA has sponsored the list since 1991. The ministry supplies Christians around the globe often worshiping in secret for fear of reprisals with basic necessities and biblical material. The list is compiled through questionnaires to field staff and other experts, said Open Doors advocacy director Lindsay Vessey. She said that Open Doors has people in 46 of the 50 countries on the list and distinguished it from State Department statistics which focus on persecution of all faiths. While North Korea remaining on top of the list may not be surprising because of the well-known Stalinist oppression practiced by dictator King Jong-Il's regime, Vessey said risks for Christians worshipping in secret continue to deteriorate. She said government attempts to restrict movement by assigned specific jobs or workforces to its citizens prompted the ministry to stop its shipments of food and Christian literature into the country. "There were two of these (work detail programs) one right after the other and our people there asked us to continue in faith (with shipments)," she said. "Christians continue to meet in secret with the threat of being placed in a political prison or labor camps." Also in the top 10 are three countries with a high value of interest to American foreign policy. Iran is second on the list amid the ongoing unrest since the heavily disputed re-election of president Mahmoud Ahmadinejad last summer. Street battles between demonstrators demanding more openness and government forces have ebbed and flowed for months. "With all the news that has come out of Iran in the past year there has been no focus on the fact that there has been a real crackdown on Christians during that time," Vessey said. Afghanistan, where 30,000 more U.S. troops are scheduled to be deployed this year, is sixth. Yemen is seventh. The country has received a great deal of attention since it is believed to have been the site of the terrorist training ground for suspected Nigerian bomber Umar Abdulmutallab who attempted to destroy a plane on a Detroit runway Christmas Day. Both Afghanistan and Yemen operate under the tents of Islamic Sharia Law which prohibits the practice of any other religions. "There has been a lot of very good information compiled by the State Department on religious freedom, but it hasn't been taken as seriously as it should," Vessey said. She said Christians interested in influencing public policy should engage elected officials, including President Obama, who has yet to appoint an at-large ambassador on religious freedom to the State Department. She also encourages concerned Christians can also become involved in advocacy against restrictive legislation in other countries to protect other overseas. Vessey and Open Doors CEO Carl Moeller said the most sought after commodity by persecuted Christians is prayer and the knowledge that their struggles haven't been forgotten. "There is a strong group of Christians in North Korea and actually the number of Christians in North Korea has grown in the last 10 years,""Moeller said in a press release. "Many are coming to Christ in the Muslim world. But we need to continue to embrace them in prayer in 2010." For a detailed look at the list, see: http://members.opendoorsusa.org/site/PageNavigator/World_Watch_List_2010
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Peter Elliott is a veteran news and sports journalist. He enjoys interviewing others about how God works in their lives and sharing that with readers. He is also a lifelong, long-suffering Chicago Cubs fan. He resides in Indianapolis with his wife and three sons.

redpajama_set_name: RedPajamaCommonCrawl


text:
Fortnite fury over how Google handled its security hole
It feels like only yesterday that we were all talking about how Epic Games was putting Android users at risk by refusing to put its hit game Fortnite into the Google Play store, and instead recommending players disable a security setting to install it.
It wasn't yesterday though. It was about three weeks ago.
Sadly, in completely predictable news, Google revealed publicly on Friday that it had discovered that Fortnite's unorthodox Android installer was vulnerable to being hijacked by other apps, allowing any app on your phone to silently download and install anything they like, including apps with full permissions to spy upon you.
Google actually privately reported this issue to Epic Games back on 15 August, and – to its credit – the makers of Fortnite worked "around the clock" confirming the vulnerability, developing a fix, testing it, and rolling it out.
However, Epic Games is not at all happy with Google's security researchers now making details of the vulnerability public.
In the opinion of Epic Games boss Tim Sweeney, Google should have waited longer before going public with details of the "man-in-the-disk" security hole.
We asked Google to hold the disclosure until the update was more widely installed. They refused, creating an unnecessary risk for Android users in order to score cheap PR points.
— Tim Sweeney (@TimSweeneyEpic) August 25, 2018
Epic Games doesn't want its game to be in the Google Play store because they don't like to give Google such a big percentage of the game's revenue.
Epic Games announces that it will release Fortnite outside of the Google Play store, and devises its own installer (which on many devices, requires for a security setting to be disabled).
Legions of security-savvy folks say that this is bad from the security point of view. I chimed in with my own article and discussed the issue on a recent edition of the "Smashing Security" podcast.
Epic Games shoots itself in the foot, by releasing an insecure Android installer for Fortnite.
Google, at no charge to Epic Games, puts some of its most skilful security researchers to work – and uncovers a serious security hole in the Fortnite installer for Android. They tell Epic Games about it.
Epic Games fixes the vulnerability, and asks Google to keep quiet about it for three months.
Google says 'no dice', and goes public about Epic Games's failure after a week.
Epic Games gets upset.
In the past I've been critical of how keen Google is to make public details of vulnerabilities in other vendors' software, which has sometimes occurred before patches are available.
Microsoft too has, in the past, been left fuming after Google disclosed details of unpatched zero-day vulnerabilities in Windows before there has been a decent amount of time to fix them and roll out the fixes to vulnerable users.
I can understand why some may feel that Google has acted inappropriately again on this occasion. But lets not forget some key points:
It was Epic Games which decided not to distribute its software in the (safer) Google Play store against the advice of security experts. It was Epic Games which failed to properly quality control one of the world's most popular video games and allow its vulnerable code to be installed on tens of millions of devices.
If Google hadn't found the security hole there is a chance that a malicious hacker would have done, and potentially could have put a large number of Android users at risk because of Epic Games's utter failure to do its job properly.
One side note:
Google's self-written vulnerability disclosure rules dictate that it will publicly reveal details of bugs 90 days after reporting them to software developers if they have not been addressed. But if a patch has been made "broadly available" then it will only wait one week before releasing details.
When Epic Games said it had released a patch, Google started the clock for disclosure.
However, Epic Games boss Tim Sweeney argues that their installer only updates itself if players runs it or runs the game.
There's a technical detail here that's important. The Fortnite installer only updates when you run it or run the game. So if a user only runs it every N days, then the update won't be installed for N days. We felt N=90 would be much safer than N=7.
In other words, Epic Games is concerned that not all installations of Fortnite will have received the update yet.
Huh. Isn't that an argument for being in the Google Play store? That way the vulnerable code would have been updated automatically rather than hopefully waiting for a user to click on an icon.
#Epic Games
#Malware
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2 Replies to "Fortnite fury over how Google handled its security hole"
not a spammer, just a sincere person says:
you're just licking google's ass
Graham Cluley says:
Google's donkey??
Anyway, if you care to read some of my numerous other articles about Google's disclosure of other software vendors' vulnerabilities you'll find your analysis is unlikely.
For instance, https://www.grahamcluley.com/google-discloses-microsoft-windows-vulnerability/