The king of geeks, Don Tillman, is surely one of the most recognisable, distinctive narrators in modern fiction. He first burst out of the genetics lab and on to the page in 2013 in The Rosie Project, in which he decides – aged 39 – that it's time to tackle the search for a compatible life partner scientifically.
To cut a long and entertaining story short, he meets a woman who fulfils none of the criteria in his 16-page questionnaire but who needs his help with a quest of her own. Against the odds, a strong mutual attraction develops, Rosie proving a perfect, intuitive mediator in the situations created by Don's social awkwardness. The book ends with wedding bells, and Don and Rosie moving from Australia to America.
When I picked up The Rosie Result – eagerly, as I'd very much enjoyed The Rosie Project – I didn't realise that this was actually the third in the series. The second, The Rosie Effect, continues on directly from the first, and details the birth and early years of Don and Rosie's son, Hudson.
However it's so easy to readjust to Don's singular view of the world that I didn't question the appearance of Hudson, fully formed, aged 10 and objecting to the family's imminent relocation back to Australia. And anyway, there's the small distracting matter of an accident with an oyster knife in the opening scene.
The unspoken implication in the first book is that Don is on the autistic spectrum. He is rigorously organised, obsessive about exercise and the productive use of time, and mathematically gifted (he has taken to estimating the BMI of every person he meets). He also had major social issues as a child, which come to light in The Rosie Result as he begins to compare his experiences and reactions with his son's.
Staff at Hudson's new school in Australia quickly identify these traits, and form the belief that all his behavioural issues will be resolved by labelling Hudson autistic and treating him accordingly. However as Don and Rosie consider the attitudes, ethics and implications of having Hudson tested, and meet people on all sides of the autism political debate, the lines become increasingly blurred.
Since Rosie is enjoying her new post and Don isn't – and because he set off a massive Twitter storm thanks to his response to a question at the end of a lecture – it's decided he will give up work to devote more time to "the Hudson problem". Don identifies numerous areas of his son's life where improvement is desirable, from learning to ride a bike and catch a ball to greeting classmates appropriately and refraining from correcting his teacher. And so, despite Hudson's initial objection to the change in parental roles, Don sets to work.
He begins to see parallels with his own father's attempts to "normalise" him as a boy, which were traumatic for him and didn't prevent a spell of severe depression caused by social isolation. Now, as a parent himself, Don can finally appreciate his own father's good intentions – while endeavouring to take a less distressing approach for Hudson.
I wasn't happy with the baddie's pro-homeopathy, anti-vaccine stance being used as evidence of stupidity, though with Don's (and evidently the author's) passionate belief in science I suppose it couldn't have been otherwise – and the refusal to consult an ophthalmologist is clearly taking things too far. In the end, the daughter herself opts to have the vaccines at an age when, as Don agrees with her anxious mother, there is "zero evidence of vaccine shock". A reasonable compromise.
The whole Rosie series is a celebration of "weird" people, and an emphatic demonstration that everyone can be happy and accepted. With the first book reviewed favourably by a parent of an autistic child, and a film of The Rosie Project currently in production, it seems Don and Hudson are set to become poster boys for those with aspie tendencies everywhere, and deservedly so.
For everyone with a geek in their life, this is a must-read; for everyone, it is uplifting, eye-opening and definite food for thought on whether we need to adapt to the world… or whether sometimes the world is a better place in accommodating us as we are.
The Rosie Result By Graeme Simsion is published by Penguin in hardback, RRP £14.99, and available from Amazon.


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Population in 2019: 3,909 (100% urban, 0% rural).
Population change since 2000: +86.8%
North Carolina median age:
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Lake Park village income, earnings, and wages data
Mean prices in 2019: all housing units: $219,115; detached houses: $229,096; townhouses or other attached units: $171,427; in 5-or-more-unit structures: $126,277
Median gross rent in 2019: $1,728.
March 2019 cost of living index in Lake Park: 94.1 (less than average, U.S. average is 100)
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3,24482.3%White alone
1173.0%American Indian alone
912.3%Two or more races
882.2%Asian alone
Races in Lake Park detailed stats: ancestries, foreign born residents, place of birth
Recent posts about Lake Park, North Carolina on our local forum with over 2,200,000 registered users. Lake Park is mentioned 419 times on our forum:
NC's 25 Largest Cities (623 replies)
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Latest news from Lake Park, NC collected exclusively by city-data.com from local newspapers, TV, and radio stations
Ancestries: American (10.7%), Russian (8.0%), English (6.9%), Irish (4.1%), British (3.3%), French Canadian (3.2%).
Land area: 0.80 square miles.
Population density: 4,912 people per square mile (average).
520 residents are foreign born (9.4% Europe, 2.9% Latin America).
This village:
North Carolina:
Nearest city with pop. 50,000+: Charlotte, NC (13.9 miles , pop. 540,828).
Nearest city with pop. 1,000,000+: Philadelphia, PA (453.8 miles , pop. 1,517,550).
Hemby Bridge, NC (1.2 miles ),
Indian Trail, NC (1.3 miles ),
Stallings, NC (1.7 miles ),
Matthews, NC (2.3 miles ),
Wesley Chapel, NC (2.4 miles ),
Unionville, NC (2.5 miles ),
Mint Hill, NC (2.6 miles ),
Weddington, NC (2.7 miles )
Area code commonly used in this area: 704
Population change in the 1990s: +1,998 (+2103.2%).
Most common industries in Lake Park, NC (%)
Finance & insurance (12.9%)
Educational services (15.9%)
Social assistance (5.3%)
Food & beverage stores (3.9%)
Most common occupations in Lake Park, NC (%)
Sales representatives, services, wholesale and manufacturing (5.5%)
Computer specialists (3.9%)
Operations specialties managers, except financial managers (3.8%)
Preschool, kindergarten, elementary, and middle school teachers (10.2%)
Average climate in Lake Park, North Carolina
AQI CO NO2 SO2 Ozone PM10 PM2.5 Pb
Carbon Monoxide (CO) [ppm] level in 2018 was 0.236. This is about average. Closest monitor was 11.1 miles away from the city center.
Nitrogen Dioxide (NO2) [ppb] level in 2018 was 6.30. This is about average. Closest monitor was 6.5 miles away from the city center.
Sulfur Dioxide (SO2) [ppb] level in 2018 was 0.166. This is significantly better than average. Closest monitor was 6.5 miles away from the city center.
Ozone [ppb] level in 2018 was 27.7. This is better than average. Closest monitor was 9.3 miles away from the city center.
Particulate Matter (PM10) [µg/m3] level in 2018 was 15.5. This is better than average. Closest monitor was 13.7 miles away from the city center.
Particulate Matter (PM2.5) [µg/m3] level in 2018 was 8.04. This is about average. Closest monitor was 12.6 miles away from the city center.
Lead (Pb) [µg/m3] level in 2007 was 0.00085. This is significantly better than average. Closest monitor was 14.9 miles away from the city center.
Lake Park-area historical tornado activity is slightly above North Carolina state average. It is 17% greater than the overall U.S. average.
On 5/5/1989, a category F4 (max. wind speeds 207-260 mph) tornado 4.9 miles away from the Lake Park village center killed one person and injured 6 people and caused between $5,000,000 and $50,000,000 in damages.
On 10/18/1990, a category F3 (max. wind speeds 158-206 mph) tornado 7.7 miles away from the village center injured 2 people and caused between $50,000 and $500,000 in damages.
Lake Park-area historical earthquake activity is significantly above North Carolina state average. It is 26% smaller than the overall U.S. average.
On 8/9/2020 at 12:07:37, a magnitude 5.1 (5.1 MW, Depth: 4.7 mi, Class: Moderate, Intensity: VI - VII) earthquake occurred 99.5 miles away from Lake Park center
On 8/23/2011 at 17:51:04, a magnitude 5.8 (5.8 MW, Depth: 3.7 mi) earthquake occurred 246.3 miles away from Lake Park center
On 8/2/1974 at 08:52:09, a magnitude 4.9 (4.3 MB, 4.9 LG, Class: Light, Intensity: IV - V) earthquake occurred 135.0 miles away from the city center
On 11/22/1974 at 05:25:55, a magnitude 4.7 (4.7 MB) earthquake occurred 153.7 miles away from the city center
On 7/27/1980 at 18:52:21, a magnitude 5.2 (5.1 MB, 4.7 MS, 5.0 UK, 5.2 UK) earthquake occurred 283.3 miles away from the city center
On 11/30/1973 at 07:48:41, a magnitude 4.7 (4.7 MB, 4.6 ML) earthquake occurred 193.9 miles away from the city center
The number of natural disasters in Union County (12) is smaller than the US average (15).
Causes of natural disasters: Hurricanes: 5, Blizzard: 1, Drought: 1, Ice Storm: 1, Tornado: 1, Tropical Storm: 1, Winter Storm: 1, Other: 1 (Note: some incidents may be assigned to more than one category).
Hospitals and medical centers near Lake Park:
LAKE PARK NURSING AND REHABILITATION CENTER (Nursing Home, about 1 miles away; INDIAN TRAIL, NC)
WOODBEND GROUP HOME (Hospital, about 6 miles away; MATTHEWS, NC)
CARRINGTON PLACE (Nursing Home, about 6 miles away; MATTHEWS, NC)
RIDGEFIELD HOME (Hospital, about 7 miles away; MONROE, NC)
KAREN LANE HOME (Hospital, about 8 miles away; MONROE, NC)
UNION REGIONAL HOME CARE (Home Health Center, about 9 miles away; MONROE, NC)
HOSPICE OF UNION COUNTY (Hospital, about 9 miles away; MONROE, NC)
Amtrak station near Lake Park:
16 miles: CHARLOTTE (1914 N. TYRON ST.) . Services: ticket office, partially wheelchair accessible, enclosed waiting area, public restrooms, public payphones, vending machines, free short-term parking, free long-term parking, call for car rental service, taxi stand, public transit connection.
Local government website: www.lakeparknc.com
Colleges/universities with over 2000 students nearest to Lake Park:
Wingate University (about 14 miles; Wingate, NC; Full-time enrollment: 2,125)
Queens University of Charlotte (about 14 miles; Charlotte, NC; FT enrollment: 2,083)
Central Piedmont Community College (about 15 miles; Charlotte, NC; FT enrollment: 15,371)
Johnson & Wales University-Charlotte (about 16 miles; Charlotte, NC; FT enrollment: 2,518)
University of North Carolina at Charlotte (about 17 miles; Charlotte, NC; FT enrollment: 22,807)
York Technical College (about 23 miles; Rock Hill, SC; FT enrollment: 3,572)
Winthrop University (about 25 miles; Rock Hill, SC; FT enrollment: 5,575)
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Union County has a predicted average indoor radon screening level less than 2 pCi/L (pico curies per liter) - Low Potential
For population 15 years and over in Lake Park:
North Carolina average:
Political contributions by individuals in Lake Park, NC
Religion statistics for Lake Park, NC (based on Union County data)
Mainline Protestant 23,392 56
Black Protestant 4,364 23
Other 1,222 5
Union County:
More about Health and Nutrition of Lake Park, NC Residents
Other Government Administration 1 $6,270 $75,240 7 $1,598
Financial Administration 0 $0 1 $1,386
Totals for Government 1 $6,270 $75,240 8 $2,984
Lake Park government finances - Expenditure in 2018 (per resident):
Current Operations - Parks and Recreation: $334,000 ($85.44)
Solid Waste Management: $185,000 ($47.33)
Regular Highways: $126,000 ($32.23)
Central Staff Services: $85,000 ($21.74)
General - Other: $42,000 ($10.74)
Financial Administration: $38,000 ($9.72)
Sewerage: $35,000 ($8.95)
Housing and Community Development: $21,000 ($5.37)
Judicial and Legal Services: $13,000 ($3.33)
Other Capital Outlay - General - Other: $35,000 ($8.95)
Lake Park government finances - Revenue in 2018 (per resident):
Charges - Parks and Recreation: $82,000 ($20.98)
Sewerage: $44,000 ($11.26)
Miscellaneous - General Revenue - Other: $5,000 ($1.28)
Interest Earnings: $0 ($0.00)
State Intergovernmental - General Local Government Support: $336,000 ($85.96)
Highways: $97,000 ($24.81)
Tax - Property: $644,000 ($164.75)
Other License: $2,000 ($0.51)
from Mecklenburg County, NC 3.15% ($87,123 average AGI)
from Cabarrus County, NC 0.17% ($59,832)
from Anson County, NC 0.14% ($28,538)
0.78% of residents moved to foreign countries ($6,057 average AGI)
to Mecklenburg County, NC 2.37% ($66,457 average AGI)
to Lancaster County, SC 0.27% ($72,077)
to Cabarrus County, NC 0.23% ($49,434)
Strongest AM radio stations in Lake Park:
WNOW (1030 AM; daytime; 9 kW; MINT HILL, NC; Owner: CHARLOTTE METRO RADIO, INC.)
WBT (1110 AM; 50 kW; CHARLOTTE, NC; Owner: JEFFERSON-PILOT COMMUNICATNS CO. OF NORTH CAROLINA)
WAGL (1560 AM; daytime; 50 kW; LANCASTER, SC; Owner: PALMETTO BROADCASTING SYSTEM, INC.)
WDEX (1430 AM; 3 kW; MONROE, NC; Owner: NEW LIFE COMMUNITY TEMPLE OF FAITH, INC.)
WIXE (1190 AM; daytime; 3 kW; MONROE, NC; Owner: MONROE BROADCASTING COMPANY, INC.)
WGFY (1480 AM; 10 kW; CHARLOTTE, NC; Owner: ABC, INC.)
WFNZ (610 AM; 5 kW; CHARLOTTE, NC; Owner: INFINITY RADIO SUBSIDIARY OPERATIONS INC.)
WCEO (840 AM; daytime; 50 kW; COLUMBIA, SC; Owner: EASTERN BROADCASTING GROUP, INC.)
WLTC (1370 AM; 20 kW; GASTONIA, NC)
WYFQ (930 AM; 5 kW; CHARLOTTE, NC; Owner: BIBLE BROADCASTING NETWORK, INC.)
WHKY (1290 AM; 50 kW; HICKORY, NC; Owner: LONG COMMUNICATIONS, LLC.)
WCGC (1270 AM; 10 kW; BELMONT, NC; Owner: WHVN, INC.)
WBAJ (890 AM; daytime; 50 kW; BLYTHEWOOD, SC; Owner: FAMILY FIRST)
Strongest FM radio stations in Lake Park:
WRCM (91.9 FM; WINGATE, NC; Owner: COLUMBIA BIBLE COLLEGE BROADCASTING CO)
WSSS (104.7 FM; CHARLOTTE, NC; Owner: INFINITY RADIO SUBSIDIARY OPERATIONS INC.)
WSOC-FM (103.7 FM; CHARLOTTE, NC; Owner: INFINITY RADIO SUBSIDIARY OPERATIONS INC.)
WWMG (96.1 FM; SHELBY, NC; Owner: CLEAR CHANNEL BROADCASTING LICENSES, INC.)
WKKT (96.9 FM; STATESVILLE, NC; Owner: CAPSTAR TX LIMITED PARTNERSHIP)
WRFX-FM (99.7 FM; KANNAPOLIS, NC; Owner: CAPSTAR TX LIMITED PARTNERSHIP)
WFAE (90.7 FM; CHARLOTTE, NC; Owner: UNIVERSITY RADIO FOUNDATION, INC.)
WLNK (107.9 FM; CHARLOTTE, NC; Owner: JEFFERSON-PILOT COMMUNICATIONS COMPANY OF NORTH CAROLINA)
WPEG (97.9 FM; CONCORD, NC; Owner: INFINITY RADIO SUBSIDIARY OPERATIONS INC.)
WNKS (95.1 FM; CHARLOTTE, NC; Owner: INFINITY RADIO SUBSIDIARY OPERATIONS INC.)
WLYT (102.9 FM; HICKORY, NC; Owner: CAPSTAR TX LIMITED PARTNERSHIP)
WBAV-FM (101.9 FM; GASTONIA, NC; Owner: INFINITY RADIO SUBSIDIARY OPERATIONS INC.)
W264AF (100.7 FM; CHARLOTTE, NC; Owner: ISOTHERMAL COMMUNITY COLLEGE)
WYFQ-FM (93.5 FM; WADESBORO, NC; Owner: BIBLE BROADCASTING NETWORK, INC.)
WTHZ (94.1 FM; LEXINGTON, NC; Owner: DAVIDSON COUNTY BROADCASTING CO, INC)
WDAV (89.9 FM; DAVIDSON, NC; Owner: THE TRUSTEES OF DAVIDSON COLLEGE)
WCHH (92.7 FM; HARRISBURG, NC; Owner: RADIO ONE OF NORTH CAROLINA, LLC)
WXRC (95.7 FM; HICKORY, NC; Owner: PACIFIC BROADCASTING GROUP, INC.)
WFMX (105.7 FM; STATESVILLE, NC; Owner: MERCURY BROADCASTING COMPANY, INC.)
WSGE (91.7 FM; DALLAS, NC; Owner: GASTON COLLEGE)
TV broadcast stations around Lake Park:
WGTB-LP (Channel 28; CHARLOTTE, NC; Owner: VICTORY CHRISTIAN CENTER, INC.)
WAFF (Channel 48; HUNTSVILLE, AL; Owner: RAYCOM AMERICA, INC.)
WSOC-TV (Channel 9; CHARLOTTE, NC; Owner: WSOC-TV HOLDINGS, INC.)
WTVI (Channel 42; CHARLOTTE, NC; Owner: CHARLOTTE-MECKLENBURG PUBLIC BROADCASTING AUTHORITY)
WCCB (Channel 18; CHARLOTTE, NC; Owner: NORTH CAROLINA BROADCASTING PARTNERS)
W25BR (Channel 25; CHARLOTTE, NC; Owner: WORD OF GOD FELLOWSHIP, INC.)
WUNG-TV (Channel 58; CONCORD, NC; Owner: UNIVERSITY OF NORTH CAROLINA)
W52CW (Channel 52; CHARLOTTE, NC; Owner: TRINITY BROADCASTING NETWORK)
WBTV (Channel 3; CHARLOTTE, NC; Owner: JEFFERSON-PILOT COMMUNICATIONS/WBTV, INC.)
WCNC-TV (Channel 36; CHARLOTTE, NC; Owner: WCNC-TV, INC.)
WJZY (Channel 46; BELMONT, NC; Owner: WJZY-TV, INC.)
WWWB (Channel 55; ROCK HILL, SC; Owner: WWWB-TV, INC.)
WAXN (Channel 64; KANNAPOLIS, NC; Owner: WSOC-TV HOLDINGS, INC.)
W34BN (Channel 34; CHARLOTTE, NC; Owner: THREE ANGELS BROADCASTING NETWORK)
WNSC-TV (Channel 30; ROCK HILL, SC; Owner: SOUTH CAROLINA EDUCATIONAL TV COMMISSION)
6.5%Electricity
Lake Park compared to North Carolina state average:
Median age below state average.
Foreign-born population percentage significantly above state average.
Number of rooms per house above state average.
Institutionalized population percentage above state average.
Number of college students above state average.
Lake Park on our top lists:
#81 on the list of "Top 101 cities with the most residents born in Ukraine (population 500+)"
#3 on the list of "Top 101 counties with the largest decrease in the number of births per 1000 residents 2000-2006 to 2007-2013 (pop 50,000+)"
#20 on the list of "Top 101 counties with the lowest ground withdrawal of fresh water for public supply (pop. 50,000+)"
#47 on the list of "Top 101 counties with the highest percentage of residents that keep firearms around their homes"
#70 on the list of "Top 101 counties with the lowest percentage of residents that smoked 100+ cigarettes in their lives"
There are no pilots and 1 other airmen in this city.
Benjamin L. Gatti (6)
Roland M. Ignacio (1)
Total of 7 patent applications in 2008-2023.

redpajama_set_name: RedPajamaCommonCrawl


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text:
\section{Introduction}
In the ultracold Bose-Einstein condensate (BEC), the many-body interaction modifies the system behaviors deviate from the ideal gas \cite{Stringari1999RMP}, which provides a platform to explore exotic quantum effects such as low-energy excitations \cite{Wieman1996PRL, Ketterle1996PRL, Davison2005RMPexcitation}, phase transitions in optical lattices \cite{Bloch2008RMPlattice, Nori2014RMPsimulation, Eckardt2017RMPlattice}, matter-wave interferometry \cite{Phillips2000PRLphase, Schmiedmayer2009RMPinterferometry}, artificial gauge potential \cite{Spielman2011NATUREsoc, Dalibard2011RMPgauge}, low-dimension physics \cite{Weiss2006NATUREcradle, Giamarchi2011RMPonedimension}, and many others. Due to the small \textit{in-situ} size of the condensate in the trap, the cold atomic sample is usually probed after certain free expansion time \cite{Ketterle1999Arxiv}. Many-body interaction plays an important role to determine the expansion dynamics. After the condensate being released from the trap, the interaction energy is converted to the kinetic energy and the initial acceleration after switch-off of the trap is determined by the gradient of the interaction energy\cite{Stringari1996PRA, Baym1996PRL, Castin1996PRL}. The expansion behaviors of different directions are dependent on the configuration of the external trap. Previously the condensate is mostly produced in an anistotropic trap (i.e., $\epsilon\neq1$, where $\epsilon$ is the aspect ratio between the axial and radial frequencies) due to the technical challenge, which leads to an increased degree of complexity in the study of the expansion behavior. In this case the expansion behaviors couldn't be analytically solved without approximation because different coupled second-order differential equations are relevant \cite{Stringari1999RMP, Castin1996PRL, Cooper1997PRL, Pitaevskii1997PLAexpansion}. Obtaining the quantum system with an analytic solution allows a more lucid description of the condensate dynamics and an immediate comparison between experiment and theory. Spherical Bose condensate in an isotropic trap (i.e., $\epsilon=1$) is a special case that the expansion behavior can be analytically solved. Here we only need to solve one differential equation due to the spherical symmetry. Nevertheless, the experimental study of this expansion is still lacking.

The spherical condensate has unique features of the many-body interaction. The excitation spectrum of the condensate is simplified by degeneracy and becomes theoretically tractable \cite{Stringari1996PRL, Stringari1999RMP, Pitaevskii1999PRA}. The reduced availability of states in a spherical trap, caused by degeneracy, has a major effect on the Landau damping rate. Quantitative calculations of this process currently have been carried out \cite{Pitaevskii1997PLA, Pitaevskii1999PRA, Burnett2000PRLsphericalexcitation, Cornell2016PRA}. Accurately measuring the collective mode of the spherical condensate in the finite-temperature regime can extract subtle many-body effects like thermal and quantum fluctuation \cite{Hu2004PRAthermal, Griffin2001PRAthermal, Zaremba2002PRAthermal, Giorgini2000PRAthermal}.

Previously Hodby et al were able to modify the aspect ratio ($\epsilon=2.83 \sim 1.6$) in a magnetic trap while keeping the confinement tight \cite{Foot2000JPBtrap}. However they do not report achieving a fully isotropic trap. Lobser and his colleagues realized an isotropic magnetic trap with the aid of the gravity force \cite{Cornell2015NaturePhy, Cornell2016PRA}. But the weak confinement (the trapping frequency $\omega \approx 2\pi\times9$ Hz) in their work is disadvantageous to obtain the pure condensate.

In this paper we produce a nearly spherical rubidium condensate in an optical dipole trap with an asphericity of 0.037. The large trapping frequency ($2\pi\times77.5$ Hz) is favorite to produce a pure condensate. Then we measure the condensate widths during the expansion process as well as the interaction energy of the condensate in the trap. The experimental results are consistent with the theoretical predictions based on the spherical symmetry. We explore the expansion dynamic process in which the the interaction energy is converted into the kinetic energy and becomes smaller versus the expansion time.

The paper is organized as follows. We first present the production of a spherical rubidium condensate in Sec. II. Then we introduce the expansion behaviors of the condensate in Sec. III. Subsequently the interaction energy of the condensate in the trap is obtained in Sec. IV. Finally, the conclusions are summarized in Sec. V.

\section{Production of a spherical Bose condensate}
The experimental configuration is composed of double magneto-optical traps (MOTs), which is similar to our previous works \cite{Jiang2016CPLbec, Jiang2018CPLplug}. $^{87}$Rb atoms are cooled and trapped in the first MOT and then transferred to the second MOT with a series of optical pushing pulses. In the second MOT the atom number is $8.5(9)\times10^8$ and the atom temperature is $320(40)$ $\mu$K. The atom temperature is reduced to 130(20) $\mu$K after a decompressed MOT process. Then the atoms are loaded into a magnetic trap by scanning the magnetic gradient to 336.0 G/cm with a period of 300 ms. Here the atom number is $2.0(7)\times10^8$ and the atom temperature is 210(25) $\mu$K. The atoms are cooled with the ratio frequency (RF) induced evaporation cooling to 15(3) $\mu K$ and subsequently transferred into a hybrid trap composed of magnetic and optical dipole fields \cite{Spielman2009PRA}. Finally we transfer cold atoms into an optical dipole trap by gradually decreasing the magnetic trap.

We produce a spherical $^{87}$Rb BEC in an optical dipole trap in which the trapping frequencies along $x,y,z-$directions are the same. As shown in Fig.~\ref{Fig1}(a), the optical dipole trap is composed of two far red-detuned laser beams with the wavelength $\lambda=1064$ nm. The Rayleigh length, $z_{R}=\pi w_{0}^{2}/\lambda$, is much larger than the beam waist $w_{0}$. For a single laser beam, the trapping frequency in the radial direction is about 200 times larger than that in the axial direction. So the trapping effect along the propagation direction can be neglected. To produce a fully isotropic trap, the gravity force should be included \cite{Cornell2015NaturePhy, Foot2000JPBtrap}. The trapping potential, which is composed of the optical dipole trap and the gravity, is given by

\begin{equation} \label{eq:trap}
\begin{split}
U \left(x,y,z\right) =& -U_{1}\exp\left(-\frac{2x^{2}}{w_{1x}^{2}}-\frac{2z^{2}}{w_{1z}^{2}}\right)\\
&-U_{2}\exp\left(-\frac{2y^{2}}{w_{2y}^{2}}-\frac{2z^{2}}{w_{2z}^{2}}\right)-mgz.
\end{split}
\end{equation}

\noindent $w_{1x}$ ($w_{2y}$) and $w_{1z}$ ($w_{2z}$) are the waists of the optical beam along the $y$ ($x$) direction, and $U_1$ and $U_2$ are the peak potential energies of the two beams, respectively. By expanding Eq. (\ref{eq:trap}) in the potential minimum $(0,0,z_{0})$ to the second order, forming a spherical BEC should satisfy the conditions

\begin{eqnarray}
U_{1}=\frac{mgw_{1x}^{2}}{2a}\sqrt{\frac{w_{1x}^{2}/w_{1z}^{4}+w_{2y}^{2}/w_{2z}^{4}}{a-1}} \label{eq:trap1}\\
U_{2}=\frac{mgw_{2y}^{2}}{2a}\sqrt{\frac{w_{1x}^{2}/w_{1z}^{4}+w_{2y}^{2}/w_{2z}^{4}}{a-1}} \label{eq:trap2}
\end{eqnarray}

\noindent where $a=w_{1x}^{2}/w_{1z}^{2}+w_{2y}^{2}/w_{2z}^{2}$. In the experiment, we can accurately adjust the intensities of the two beams to simultaneously match Eq. (\ref{eq:trap1}) and Eq. (\ref{eq:trap2}).

\begin{figure}
\centerline{\includegraphics[width=8.5cm]{Fig1-OscillationFrequency.pdf}}
\caption{(color online) (a) Experimental setup. The optical dipole trap is composed of two focused red-detuned laser beams in $x$ and $y$ directions. The gravity is in the $-z$ direction. Ultracold atoms are simultaneously probed in the vertical and horizontal directions. (b) Measuring the trapping frequencies by probing the oscillations of the centers of mass along three directions, respectively. Each experimental data is the average of three measurements. The solid line is the fitting with a sine wavefunction.}  \label{Fig1}
\end{figure}

By displacing atoms away from the equilibrium position for 2 ms, we measure the oscillation of the center of the mass (COM) of the atomic cloud in the trap along the $x$, $y$ and $z$ directions, respectively. The experimental results are shown in Fig.~\ref{Fig1}(b). Using a sine wavefunction $r_{i}=A_{i}\sin(\omega_{i}t+\phi_{i})$ ($i=x, y, z$ and $r_{i}\rightarrow i$) to fit the experimental data, we get the trapping frequencies: $\omega_{x}= 2\pi\times 76.7(14)$ Hz, $\omega_{y}= 2\pi\times 76.5(6)$ Hz, $\omega_{z}= 2\pi\times 79.4(12)$ Hz. The frequency uncertainties in the parenthesis are from the fitting process. So the mean trapping frequency $\bar{\omega}= \left(\omega_{x}+\omega_{y}+\omega_{z}\right)/3 = 2\pi\times 77.5$ Hz which is much larger than that in the reference \cite{Cornell2015NaturePhy}. The asphericity $A=(\omega_{max}-\omega_{min})/\bar{\omega}\approx 0.037$, where $\omega_{max}$, $\omega_{min}$ are the maximum and minimum trapping frequencies along three directions, respectively. The tight confinement here is favorite to produce a pure Bose condensate with negligible thermal gases and obtain experimental data with a high signal-to-noise ratio. We improve the position stability of the optical trap beam better than 3 $\mu$m to achieve a stable spherical BEC. The atoms stay in the spin state $|F, m_{F}\rangle = |1, -1\rangle$. The atom number is about $1.2\times10^5$. The BEC is well in the hydrodynamic limit with the adimensional parameter $N a_s/a_{ho}\approx 570\gg 1$~\cite{Stringari1996PRL}, where $a_s$ is the s-wave scattering length, $a_{ho}=\sqrt{\hbar/m \omega}$ is the harmonic oscillator length of the trap, and $N$ is the atom number.

After suddenly switching off the optical trap, we measure the aspect ratio $\eta(t)$ of the condensate during the free expansion. The experimental results are shown in Fig.~\ref{Fig2}. The condensate width $R_{i}(t)$ during the expansion is obtained by fitting the optical density of the image with a Thomas-Fermi (TF) distribution along the corresponding direction. For the images probed in the horizontal direction as shown in Fig.~\ref{Fig1}(a), $\eta(t)=R_{\parallel}(t)/R_z(t)$ where $R_{\parallel}(t)$ and $R_z(t)$ are the TF radii in the horizontal and vertical directions, respectively. For the images probed in the vertical direction, $\eta(t)=R_{x}(t)/R_y(t)$ where $R_{x}(t)$ and $R_y(t)$ are the TF radii in the $x$ and $y$ directions, respectively. $\eta(t)$ remains unity during the free expansion, which is unique for a spherical BEC. For an anisotropic BEC, the expansion is anisotropic and the aspect ratio $\eta(t)$ approaches an asymptotic value dependent on the ratio of the trapping frequencies \cite{Rempe1998EPL, Rempe1998APB, Ketterle1996PRLrelease, Stringari1999RMP, Castin1996PRL}.

\begin{figure}
\centerline{\includegraphics[width=7cm]{Fig2-AspectRatio.pdf}}
\caption{(Color online) Isotropic expansion of the condensate. The upper row shows exemplary images probed in the vertical direction for five expansion times. The lower row shows the aspect ratio $\eta(t)$ versus the expansion time $t$. The black squares (red circles) are for the images probed in the horizontal (vertical) direction. Each error bar indicates the uncertainty of three measurements. The dashed line denotes the value of unity. }
\label{Fig2}
\end{figure}

\section{Expansion behavior of the condensate}

Many-body interaction plays an important role to determine the expansion dynamics of BEC. Fig.~\ref{Fig3}(a) briefly indicates variations of different energy components during the expansion process \cite{Stringari1999RMP, Stringari1996PRA, Cooper1997PRL}. The chemical potential $\mu=E_{kin}+E_{p}+2E_{int}$ is composed of kinetic energy $E_{kin}$, potential energy $E_{p}$ and interaction energy $E_{int}$. In the trap, $E_{kin}$ is negligibly small and $E_{p}=1.5E_{int}$ according to the Virial relation $2E_{kin}-2E_{p}+3E_{int}=0$. After BEC being released from the trap, $E_{p}$ is switched off and $E_{int}$ starts to be converted into $E_{kin}$ gradually, which makes the release energy $E_{rel}=E_{kin}+E_{int}$ keep constant during the expansion. After a long-time expansion, the interaction energy is completely converted to the kinetic energy. This provides a efficient way to measure the interaction energy of BEC in the trap by probing the expansion velocity in the long-time expansion, which will be followed in section IV.

\begin{figure}
\centerline{\includegraphics[width=8.5cm]{Fig3-ExpansionScheme.pdf}}
\caption{(Color online) (a) Schematics of energy components during the free expansion. The trapping potential is switched off at $t=0$. $E_{p}$ is the potential energy, $E_{rel}$ is the release energy, $E_{kin}$ is the kinetic energy, $E_{int}$ is the interaction energy and $\mu$ is the chemical potential. After a long-time expansion ($t>t_{1}$), interaction energy is completely converted to kinetic energy. (b) The scaling factor $\lambda(\tau)=R(\tau)/R(0)$ versus the scaling expansion time $\tau$. $R(0)$ is the TF radius of the BEC in the trap and $\tau = \omega t$. The black solid curve denotes the calculation with Eq.(\ref{eq:exp11}) for the whole expansion process. The red dashed curve indicates the calculation with Eq.(\ref{eq:exp2}) for the short-time expansion. The blue dotted curve is the calculation with Eq.(\ref{eq:exp3}) for the long-time expansion . }\label{Fig3}
\end{figure}

During the expansion, the atomic cloud experiences just a free dilatation. Three scaling factors, i.e., $\lambda_{i}\left(t\right)\equiv r_{i}\left(t\right)/r_{i}\left(0\right)$($i=x,y,z$), may be introduced as in references \cite{Stringari1999RMP, Castin1996PRL, Rempe1998APB, Rempe1998EPL}, which describe the trajectory of any infinitesimally small fraction at the position ${\bf{r}}\left(t\right)$ of the moving cloud. For an axially symmetric BEC mostly produced previously (i.e., $\lambda_{x}(t)=\lambda_{y}(t)\neq \lambda_{z}(t)$), it is required to solve two coupled second-order differential equations to get the evolution of the scaling factors \cite{Stringari1999RMP, Castin1996PRL, Cooper1997PRL, Pitaevskii1997PLAexpansion}. In this case analytic solution is generally absent. Here for the spherical BEC with $\lambda(t)=\lambda_{x}(t)=\lambda_{y}(t)= \lambda_{z}(t)$, the previous two coupled differential equations simply merge into one,
\begin{equation}
\frac{\partial^{2}\lambda}{\partial \tau^{2}}=\lambda^{-4},
\end{equation}
whose solution can be obtained analytically,
\begin{equation}
\tau = -\sqrt{\frac{3}{2}}\cdot\frac{\sqrt{\pi}\Gamma(2/3)}{\Gamma(1/6)} + \sqrt{\frac{3}{2}}\lambda \cdot\, _{2}F_{1}\left(-\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{1}{\lambda^{3}}\right), \label{eq:exp11}
\end{equation}
where $\tau=\omega t$, $\Gamma(\cdot)$ is the Gamma function, and $_{2}F_{1}(a,b,c,z)$ is the hypergeometric function.

According to Eq.(\ref{eq:exp11}), we can easily obtain the asymptotic behavior of the scaling factor $\lambda\left(\tau\right)$ for a short- or long-time expansion,
\begin{eqnarray}
\lambda (\tau) \approx 1+\tau^2/2- \tau^4/6, (\tau \rightarrow 0)  \label{eq:exp2}\\
\lambda (\tau) \approx \sqrt{2/3} \tau+\frac{\sqrt{\pi}\Gamma(1/6)}{\Gamma(2/3)}. (\tau \rightarrow \infty)  \label{eq:exp3}
\end{eqnarray}

The scaling factor of the condensate during the expansion process is shown in Fig.~\ref{Fig3}(b). For the short-time expansion ($\tau\rightarrow 0$), ${\partial \lambda}/{\partial \tau} \approx \tau$ and ${\partial^2 \lambda}/{\partial \tau^2}=1-2\tau^2$. This means that after the trap being switched off, the expansion is speeding but the acceleration decreases versus the expansion time. These behaviors can be explained that the interaction energy is gradually converted into the kinetic energy and decreases versus the expansion time \cite{Ketterle1996PRLrelease, Cooper1997PRL, Stringari1999RMP}. For the long-time expansion ($\tau \rightarrow \infty$), ${\partial \lambda}/{\partial \tau} = \sqrt{2/3}$. It turns out that the interaction energy has been completely converted to the kinetic energy and the expansion velocity finally reaches a constant value. The quantitative calculations in Fig.~\ref{Fig3}(b) can clearly confirm the expansion dynamics as shown in Fig.~\ref{Fig3}(a).

In Fig. \ref{Fig4} we measure the scaling factor during the expansion, which is extracted by measuring the size of the atomic cloud after some expansion time. Due to the limited resolution of the imaging system ($\Delta r \approx 7.6$ $\mu$m) \cite{Jiang2018CPLplug}, we only show the experimental data for expansion times larger than 7 ms. Here the size of the cloud is defined as $R(\tau)=\left[R_{x}(\tau)+R_{y}(\tau)+R_{z}(\tau)\right]/3$, where $R_{i}(\tau)$ ($i=x,y,z$) is the TF radius along the principle axis, and $\lambda(\tau)=R(\tau)/R(0)$. Under the TF approximation, $R(0)$ is calculated from the atom number and the trapping frequency, and $R(\tau)$ is measured in the experiment. Experimental results at three atomic numbers are consistent with the theoretical prediction of Eq. (\ref{eq:exp11}).

\begin{figure}
\centerline{\includegraphics[width=7cm]{Fig4-ScaledFactor.pdf}}
\caption{(Color online) Scaling factor $\lambda(\tau)$  versus the scaling expansion time $\tau$. The black solid curve is the calculation with Eq.(\ref{eq:exp11}). Blue squares, red circles and green diamonds denote the measurements with atom numbers of $0.55 \times 10^5$, $0.66 \times 10^5$ and $1.40 \times 10^5$. Each error bar is the uncertainty of three measurements.}\label{Fig4}
\end{figure}

\section{Interaction energy of the condensate}
During the expansion, the interaction energy is gradually converted into the kinetic energy $E_{kin}$. After atoms being releasing from the trap, $E_{kin}$ can be written in the following integral
\begin{equation}
E_{kin}=\int\frac{1}{2}m\left[\frac{d{\bf r}\left(t\right)}{dt}\right]^{2}n\left[{\bf r}\left(t\right)\right]\mathcal{D}\left[{\bf r}\left(t\right)\right],\label{eq:4.1}
\end{equation}
 where $n\left[{\bf r}\left(t\right)\right]$ is the density at the
position ${\bf r}\left(t\right)$, and $m$ is the atomic mass. Using
${\bf r}\left(t\right)=\lambda\left(t\right){\bf r}\left(0\right)$,
Eq.(\ref{eq:4.1}) becomes
\begin{equation}
E_{kin}=\frac{1}{2}m\left[\frac{d\lambda\left(t\right)/dt}{\lambda\left(t\right)}\right]^{2}\int{\bf r}\left(t\right)^{2}n\left[{\bf r}\left(t\right)\right]\mathcal{D}\left[{\bf r}\left(t\right)\right].\label{eq:4.2}
\end{equation}
It was shown in \cite{Castin1996PRL} that the density $n\left({\bf r}\right)$ still satisfies the
generalized TF distribution for a time-dependent problem, which takes the form of
\begin{equation}
n\left[{\bf r}\left(t\right)\right]=\frac{15}{8\pi R\left(t\right)^{3}}\left[1-\frac{r\left(t\right)^{2}}{R\left(t\right)^{2}}\right],\label{eq:4.3}
\end{equation}
for a spherical atomic cloud, where $R\left(t\right)$ is the generalized TF radius at
the time $t$, $n\left[{\bf r}\left(t\right)\right]$ has been normalized
to unity, i.e., $\int_{\mathcal{S}}n\left[{\bf r}\left(t\right)\right]\mathcal{D}\left[{\bf r}\left(t\right)\right]=1$,
and $\mathcal{S}$ is a spherical domain with the radius $R\left(t\right)$.
Substituting Eq.(\ref{eq:4.3}) into (\ref{eq:4.2}), we easily obtain
\begin{equation}
E_{kin}=\frac{1}{2}m\cdot\frac{3}{7}\left[\frac{dR\left(t\right)}{dt}\right]^{2}.\label{eq:4.4}
\end{equation}

As shown in Fig. \ref{Fig3}, the interaction energy is completely converted to the kinetic energy after a long-time expansion. So the interaction energy $E_{int}$ at $t\rightarrow 0$ is roughly equivalent to the kinetic energy $E_{kin}$ at $t\rightarrow \infty$.

We can check the validity of Eq.(\ref{eq:4.4}). From Eq.(\ref{eq:exp3}) the size of the cloud $R\left(t\right)$ as $t\rightarrow\infty$ should behave as
\begin{equation}
R\left(t\right)\approx\sqrt{\frac{2}{3}}\omega tR\left(0\right),\label{eq:2.7}
\end{equation}
where $R\left(0\right)$ is the size at $t=0$. Then the interaction energy of the condensate in the trap becomes
\begin{equation}
E_{int}=\frac{1}{2}m\cdot\frac{3}{7}\cdot\frac{2}{3}\omega^{2}R\left(0\right)^{2}=\frac{2}{7}\cdot\frac{1}{2}m\omega^{2}R\left(0\right)^{2}=\frac{2}{7}\mu,\label{eq:2.8}
\end{equation}
and $\mu=m\omega^{2}R\left(0\right)^{2}/2$ is the chemical potential in the trap. Eq.(\ref{eq:2.8}) is consistent with the well-known result $E_{int}=2\mu/7$ \cite{Stringari1999RMP}.

\begin{figure}
\centerline{\includegraphics[width=7cm]{Fig5-ReleasEnergy.pdf}}
\caption{(Color online) Interaction energy of the condensate in the trap versus the atom number. The solid curve is the theoretical calculation with Eq. (\ref{eq:2.9}).  The measurements are for atom numbers of $0.55 \times 10^5$, $0.66 \times 10^5$ and $1.40 \times 10^5$. The error bar is the uncertainty in fitting the expansion velocity. The insert shows an example for linearly fitting the measured TF radii of the condensate with the atom number of $1.40 \times 10^5$. The expansion times are larger than 14 ms. }\label{Fig5}
\end{figure}

In the experiment we extract the expansion velocity for the long-time expansion by linearly fitting the TF radii of the condensate. One example of this fitting process is shown in the inset of Fig. \ref{Fig5}. The expansion times are larger than $t>14$ ms (i.e., $\tau>6.8$), which ensures that the expansion velocity has approached the constant value (see Fig.~\ref{Fig3}(b)). Then the interaction energy of the condensate in the trap can be calculated with Eq. (\ref{eq:4.4}).

On the other hand, the chemical potential of the condensate in the trap can be calculated with \cite{Stringari1996PRA, Baym1996PRL, Stringari1999RMP, Castin1996PRL}

\begin{equation}
\mu=\frac{\hbar \omega}{2}(\frac{15Na_{s}}{a_{ho}})^{2/5}.\label{eq:2.9}
\end{equation}
The interaction energy of the condensate in the trap versus the atom number is ploted  in Fig. \ref{Fig5}. The experimental measurements are consistent with the theoretical prediction with Eq. (\ref{eq:2.9}).

\section{Conclusions}
In conclusion, we experimentally observe the expansion behaviors of a spherical Bose condensate. A spherical rubidium condensate is produced in an optical dipole trap and the characteristic isotropic expansion is observed in the experiment. The condensate widths during the expansion process as well as the interaction energy of the condensate in the trap are measured. The analytic solution of the expansion behavior of the condensate is derived, as a quantitative comparison with the experimental measurements. We find that the expansion in the short time is speeding and then after a long time the expansion velocity is constant. The intrinsic mechanics of this behavior is that the interaction energy is converted into the kinetic energy at the beginning of the expansion and the kinetic energy dominates after a long-time expansion.

This work has already been accessible to study the frequency shift and Landau damping of low-energy excitations in a spherical condensate, which allows the immediate comparison between the experiment and theory due to the simplified excitation spectrum by degeneracy \cite{Stringari1996PRL, Stringari1999RMP, Pitaevskii1999PRA, Pitaevskii1997PLA, Pitaevskii1999PRA, Burnett2000PRLsphericalexcitation, Cornell2016PRA, Hu2004PRAthermal, Griffin2001PRAthermal, Zaremba2002PRAthermal, Giorgini2000PRAthermal}. Compared to a magnetic trap, the spherical trap composed of the optical field is advantageous to study the non-equilibrium dynamics with the spherical symmetry, where fast modulation of the confinement strength is required.

\section*{Acknowledgments}
This work has been supported by the NKRDP (National Key Research and Development Program) under Grant No. 2016YFA0301503, NSFC (Grant No. 11674358, 11434015, 11474315) and CAS under Grant No. YJKYYQ20170025.

redpajama_set_name: RedPajamaArXiv


text:
Tesla Engineer Joins Apple's Growing Team of Automated Car Experts
iCar is coming?
By Reuters Aug 21, 2015, 1:53pm EDT
Share All sharing options for: Tesla Engineer Joins Apple's Growing Team of Automated Car Experts
Consumer electronics company Apple has hired a senior engineer from electric car maker Tesla Motors, according to a LinkedIn posting, as part of Apple's effort to build a team of experts in automated driving. A LinkedIn profile for Jamie Carlson shows that he has left Tesla and moved to Apple. At least six others with experience developing self-driving technology and systems have joined Apple, according to their LinkedIn profiles.
Attempts to reach all seven people were unsuccessful and Apple declined to comment.
Sources have said that Apple is developing a car and studying self-driving technology, but it is unclear if the iPhone maker is designing a vehicle that could drive itself. Since January, Apple has hired Megan McClain, a former Volkswagen AG engineer with expertise in automated driving, and Vinay Palakkode, a graduate researcher at Carnegie Mellon University, a hub of automated driving research.
In August, Apple hired Xianqiao Tong, an engineer who developed computer vision software for driver assistance systems at microchip maker Nvidia.
The Wall Street Journal has reported that Apple hired Paul Furgale, former deputy director of the Autonomous Systems Lab at the Swiss Federal Institute of Technology, earlier this year.
So-called advanced driver assistance systems, or ADAS, handle tasks such as keeping a vehicle in a lane or driving by itself in stop-and-go traffic, and they are considered the building blocks for self-driving cars.
According to Carlson's LinkedIn profile, he joined Apple in August in an unnamed position in a special projects group. Through July, Carlson was an engineer on Tesla's Autopilot self-driving car program, and before that he worked on automotive vision systems for Michigan-based supplier Gentex Corp.
Other Apple hires since September 2014 with similar experience have worked at automakers BMW AG, Volkswagen and Ford, automotive suppliers Delphi Automotive, Robert Bosch GmbH and TRW, now a part of ZF Friedrichshafen AG, according to their LinkedIn profiles. Among those hired last fall were Sanjai Massey, an engineer with experience in developing connected and automated vehicles at Ford and several suppliers; Stefan Weber, a former Bosch engineer with experience in video-based driver assistance systems, and Lech Szumilas, a former Delphi research scientist with expertise in computer vision and object detection.
(Reporting by Paul Lienert and Julia Love; editing by Peter Henderson and Matthew Lewis)