Evolution of the distribution of rapidities under atom losses in the - - PowerPoint PPT Presentation

evolution of the distribution of rapidities under atom
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Evolution of the distribution of rapidities under atom losses in the - - PowerPoint PPT Presentation

Evolution of the distribution of rapidities under atom losses in the 1D Bose gas Jrme Dubail LPCT, CNRS and Universit de Lorraine, Nancy Based on arXiv:2006.03583 with: Isabelle Bouchoule (Institut dOptique, Palaiseau) Benjamin Doyon


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SLIDE 1

Evolution of the distribution of rapidities under atom losses in the 1D Bose gas

Jérôme Dubail LPCT, CNRS and Université de Lorraine, Nancy

Based on arXiv:2006.03583 with: Isabelle Bouchoule (Institut d’Optique, Palaiseau) Benjamin Doyon (King’s College, London)

1

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SLIDE 2

2

after short relaxation time , the macrostate in the box is entirely characterized by (particle density, mean velocity, mean energy per particle)

τrelax

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n, u, e

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Slow losses in a chaotic/ergodic gas (2D hard spheres)

slide-3
SLIDE 3

3

Slow losses in a chaotic/ergodic gas (2D hard spheres)

imagine that some of the particles are lost, at a rate Γ

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Γ ⌧ 1/τrelax

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n, u, e

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dn dt = −Γn

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du dt = −Γu

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de dt = −Γe

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time

as long as , the system always has time to relax. So the macrostate in the box is still characterized by , but:

loss 1 loss 2 loss 3

τrelax

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relaxed state

τrelax

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τrelax

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relaxed state relaxed state

∼ Γ−1

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∼ Γ−1

<latexit sha1_base64="rAKdnZka4WRFrqDz+eEmS+cGyxk=">AC2HicjVHLSsNAFD2N73fVpZtgEVxoSUSwy4ILXSpYFa2WSZzWoXmRmQilCO7ErT/gVr9I/AP9C+MU/CB6IQkZ86958zce4MsElJ53kvJGRoeGR0bn5icmp6ZnSvPLxzKtMhD3gjTKM2PAyZ5JBLeUEJF/DjLOYuDiB8F3W0dP7riuRpcqB6GT+LWScRbREyRVSrvNiUInaba25zh8UxO+v+9etcsWrema5P4FvQV27aXlZzRxgRQhCsTgSKAIR2CQ9JzCh4eMuDP0icsJCRPnuMYkaQvK4pTBiO3St0O7U8smtNe0qhDOiWiNyelixXSpJSXE9anuSZeGfN/ubdN576bj36B9YrJlbhkti/dIPM/+p0LQpt1EwNgmrKDKOrC61LYbqib+5+qkqRQ0acxhcUzwmHRjnos2s0tSue8tM/NVkalbvQ5tb4E3fkgbsfx/nT3C4UfW9qr+/WanX7KjHsYRlrNI8t1DHLvbQIO8eHvCIJ+fEuXFunbuPVKdkNYv4spz7d+SLlkA=</latexit><latexit sha1_base64="rAKdnZka4WRFrqDz+eEmS+cGyxk=">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</latexit><latexit sha1_base64="rAKdnZka4WRFrqDz+eEmS+cGyxk=">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</latexit><latexit sha1_base64="rAKdnZka4WRFrqDz+eEmS+cGyxk=">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</latexit>
slide-4
SLIDE 4

Slow losses in a integrable gas (1D hard spheres or hard rods)

slide-5
SLIDE 5

Slow losses in a integrable gas (1D hard spheres or hard rods)

to characterize the macrostate in the box, one needs the entire distribution of velocities:

ρ(v) = 1 L

N

X

i=1

δ(v − vi)

<latexit sha1_base64="7LqdRUAO0fiODbF953J+gEeLec=">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</latexit><latexit sha1_base64="7LqdRUAO0fiODbF953J+gEeLec=">AC/nicjVHLShxBFD12Hj4TJ2bpnAQFJKhWwTNYmAgmyCKDgq2DpU19RoMf2iunpAmgH/JLvsxG1+INu4FP9A/8JblRaSNBquvrUuecrlsV5bEqjO/fTHgvXr56PTk1PTM79+btfOPdwl6RlVrIrsjiTB9EvJCxSmXKBPLg1xLnkSx3I+Gn219fyR1obJ015zl8ijhJ6kaKMENUb3Gp1CfZmxltMrCD6xtp3CguaiCcfV1zMKiTHqVagfj4y0W9mVsOGk/jnpqlfUaTb/lu8Eeg6AGTdRjO2tcI0QfGQRKJBIYQjH4CjoOUQAHzlxR6iI04SUq0uMUPeklSFJzYIc0ntDqs2ZTWNrNwbkF/ienV5GRYJk9GOk3Y/o25eumSLfu/7Mpl2r2d0TeqsxJiDU6Jfcr3oHyuz/ZiMCm60FRT7ljbHeiTindqdidsz+6MpSQE2dxn+qasHDOh3NmzlO43u3Zcle/dUrL2rWotSXu7C7pgoN/r/Mx2FtrBX4r2Flvdjbrq57CIpawQve5gQ6+YBtdyv6Gn/iFK+/c+5deJe/pd5E7XmPv4b34x7S8qO9</latexit><latexit sha1_base64="7LqdRUAO0fiODbF953J+gEeLec=">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</latexit><latexit sha1_base64="7LqdRUAO0fiODbF953J+gEeLec=">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</latexit>

L

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ρ(v)

<latexit sha1_base64="YcVrl3854ie7xFEhzVef29pcyTE=">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</latexit><latexit sha1_base64="YcVrl3854ie7xFEhzVef29pcyTE=">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</latexit><latexit sha1_base64="YcVrl3854ie7xFEhzVef29pcyTE=">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</latexit><latexit sha1_base64="YcVrl3854ie7xFEhzVef29pcyTE=">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</latexit>

v

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v1

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v2

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vN

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v

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v1

<latexit sha1_base64="QBAjgx9j/VIP5l1Ht8XU57kGdwk=">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</latexit><latexit sha1_base64="QBAjgx9j/VIP5l1Ht8XU57kGdwk=">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</latexit><latexit sha1_base64="QBAjgx9j/VIP5l1Ht8XU57kGdwk=">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</latexit><latexit sha1_base64="QBAjgx9j/VIP5l1Ht8XU57kGdwk=">ACxnicjVHLSsNAFD2Nr/qunQTLIKrkoigy6KbLivaB9RSkum0hubFZFIpRfAH3OqniX+gf+GdcQpqEZ2Q5My595yZe6+fhkEmHe1YC0sLi2vFfX1jc2t7ZLO7vNLMkF4w2WhIlo+17GwyDmDRnIkLdTwb3ID3nLH12oeGvMRYk8bWcpLwbecM4GATMk0RdjXtur1R2Ko5e9jxwDSjDrHpSesEN+kjAkCMCRwxJOISHjJ4OXDhIietiSpwgFOg4xz3WSJtTFqcMj9gRfYe06xg2pr3yzLSa0SkhvYKUNg5Jk1CeIKxOs3U8186K/c17qj3V3Sb0941XRKzELbF/6WaZ/9WpWiQGONM1BFRTqhlVHTMue6Kurn9pSpJDilxCvcpLgzrZz12daTNeueuvp+JvOVKzaM5Ob413dkgbs/hznPGgeV1yn4l6elKvnZtRF7OMARzTPU1RQx0N8h7iEU94tmpWbOXW3WeqVTCaPXxb1sMHCUeQHw=</latexit>

v2

<latexit sha1_base64="iVJAn14qj+ZUX75oDbR5puDaV8=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LopsuK9gG1lGQ6rUPTJCSTSimCP+BWP038A/0L74xTUIvohCRnzr3nzNx7/TgQqXSc15y1tLyupZfL2xsbm3vFHf3mUJYw3WBRESdv3Uh6IkDekAFvxwn3xn7AW/7oQsVbE56kIgqv5Tm3bE3DMVAME8SdTXpVXrFklN29LIXgWtACWbVo+ILbtBHBIYMY3CEkIQDeEjp6cCFg5i4LmbEJYSEjnPco0DajLI4ZXjEjug7pF3HsCHtlWeq1YxOCehNSGnjiDQR5SWE1Wm2jmfaWbG/ec+0p7rblP6+8RoTK3FL7F+6eZ/daoWiQHOdA2Cao1o6pjxiXTXVE3t79UJckhJk7hPsUTwkwr532tSbVtavejr+pjMVq/bM5GZ4V7ekAbs/x7kImpWy65Tdy5NS9dyMOo8DHOKY5nmKmqo0HeQziCc9WzQqtzLr7TLVyRrOPb8t6+ALp5Ag</latexit><latexit sha1_base64="iVJAn14qj+ZUX75oDbR5puDaV8=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LopsuK9gG1lGQ6rUPTJCSTSimCP+BWP038A/0L74xTUIvohCRnzr3nzNx7/TgQqXSc15y1tLyupZfL2xsbm3vFHf3mUJYw3WBRESdv3Uh6IkDekAFvxwn3xn7AW/7oQsVbE56kIgqv5Tm3bE3DMVAME8SdTXpVXrFklN29LIXgWtACWbVo+ILbtBHBIYMY3CEkIQDeEjp6cCFg5i4LmbEJYSEjnPco0DajLI4ZXjEjug7pF3HsCHtlWeq1YxOCehNSGnjiDQR5SWE1Wm2jmfaWbG/ec+0p7rblP6+8RoTK3FL7F+6eZ/daoWiQHOdA2Cao1o6pjxiXTXVE3t79UJckhJk7hPsUTwkwr532tSbVtavejr+pjMVq/bM5GZ4V7ekAbs/x7kImpWy65Tdy5NS9dyMOo8DHOKY5nmKmqo0HeQziCc9WzQqtzLr7TLVyRrOPb8t6+ALp5Ag</latexit><latexit sha1_base64="iVJAn14qj+ZUX75oDbR5puDaV8=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LopsuK9gG1lGQ6rUPTJCSTSimCP+BWP038A/0L74xTUIvohCRnzr3nzNx7/TgQqXSc15y1tLyupZfL2xsbm3vFHf3mUJYw3WBRESdv3Uh6IkDekAFvxwn3xn7AW/7oQsVbE56kIgqv5Tm3bE3DMVAME8SdTXpVXrFklN29LIXgWtACWbVo+ILbtBHBIYMY3CEkIQDeEjp6cCFg5i4LmbEJYSEjnPco0DajLI4ZXjEjug7pF3HsCHtlWeq1YxOCehNSGnjiDQR5SWE1Wm2jmfaWbG/ec+0p7rblP6+8RoTK3FL7F+6eZ/daoWiQHOdA2Cao1o6pjxiXTXVE3t79UJckhJk7hPsUTwkwr532tSbVtavejr+pjMVq/bM5GZ4V7ekAbs/x7kImpWy65Tdy5NS9dyMOo8DHOKY5nmKmqo0HeQziCc9WzQqtzLr7TLVyRrOPb8t6+ALp5Ag</latexit><latexit sha1_base64="iVJAn14qj+ZUX75oDbR5puDaV8=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6LopsuK9gG1lGQ6rUPTJCSTSimCP+BWP038A/0L74xTUIvohCRnzr3nzNx7/TgQqXSc15y1tLyupZfL2xsbm3vFHf3mUJYw3WBRESdv3Uh6IkDekAFvxwn3xn7AW/7oQsVbE56kIgqv5Tm3bE3DMVAME8SdTXpVXrFklN29LIXgWtACWbVo+ILbtBHBIYMY3CEkIQDeEjp6cCFg5i4LmbEJYSEjnPco0DajLI4ZXjEjug7pF3HsCHtlWeq1YxOCehNSGnjiDQR5SWE1Wm2jmfaWbG/ec+0p7rblP6+8RoTK3FL7F+6eZ/daoWiQHOdA2Cao1o6pjxiXTXVE3t79UJckhJk7hPsUTwkwr532tSbVtavejr+pjMVq/bM5GZ4V7ekAbs/x7kImpWy65Tdy5NS9dyMOo8DHOKY5nmKmqo0HeQziCc9WzQqtzLr7TLVyRrOPb8t6+ALp5Ag</latexit>

vN

<latexit sha1_base64="+azdXGcR5olXvw3Vf8p4CiDiImE=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl05VUtA+opSTaR2aJmEyqZQi+ANu9dPEP9C/8M6YglpEJyQ5c+49Z+be68eBSJTjvOashcWl5ZX8amFtfWNzq7i90iVDJeZ1EQyZbvJTwQIa8roQLeiX3Rn7Am/7wXMebYy4TEYXahLzsgbhKIvmKeIuhp3L7rFklN2zLngZuBErJVi4ovuEPERhSjMARQhEO4CGhpw0XDmLiOpgSJwkJE+e4R4G0KWVxyvCIHdJ3QLt2xoa0156JUTM6JaBXktLGAWkiypOE9Wm2iafGWbO/eU+Np7bhP5+5jUiVuGW2L90s8z/6nQtCn2cmhoE1RQbRlfHMpfUdEXf3P5SlSKHmDiNexSXhJlRzvpsG01iate9Uz8zWRqVu9ZlpviXd+SBuz+HOc8aByVXafsXh6XKmfZqPYwz4OaZ4nqKCKGurkPcAjnvBsVa3QSq27z1Qrl2l28W1ZDx9OJ5A8</latexit><latexit sha1_base64="+azdXGcR5olXvw3Vf8p4CiDiImE=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl05VUtA+opSTaR2aJmEyqZQi+ANu9dPEP9C/8M6YglpEJyQ5c+49Z+be68eBSJTjvOashcWl5ZX8amFtfWNzq7i90iVDJeZ1EQyZbvJTwQIa8roQLeiX3Rn7Am/7wXMebYy4TEYXahLzsgbhKIvmKeIuhp3L7rFklN2zLngZuBErJVi4ovuEPERhSjMARQhEO4CGhpw0XDmLiOpgSJwkJE+e4R4G0KWVxyvCIHdJ3QLt2xoa0156JUTM6JaBXktLGAWkiypOE9Wm2iafGWbO/eU+Np7bhP5+5jUiVuGW2L90s8z/6nQtCn2cmhoE1RQbRlfHMpfUdEXf3P5SlSKHmDiNexSXhJlRzvpsG01iate9Uz8zWRqVu9ZlpviXd+SBuz+HOc8aByVXafsXh6XKmfZqPYwz4OaZ4nqKCKGurkPcAjnvBsVa3QSq27z1Qrl2l28W1ZDx9OJ5A8</latexit><latexit sha1_base64="+azdXGcR5olXvw3Vf8p4CiDiImE=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl05VUtA+opSTaR2aJmEyqZQi+ANu9dPEP9C/8M6YglpEJyQ5c+49Z+be68eBSJTjvOashcWl5ZX8amFtfWNzq7i90iVDJeZ1EQyZbvJTwQIa8roQLeiX3Rn7Am/7wXMebYy4TEYXahLzsgbhKIvmKeIuhp3L7rFklN2zLngZuBErJVi4ovuEPERhSjMARQhEO4CGhpw0XDmLiOpgSJwkJE+e4R4G0KWVxyvCIHdJ3QLt2xoa0156JUTM6JaBXktLGAWkiypOE9Wm2iafGWbO/eU+Np7bhP5+5jUiVuGW2L90s8z/6nQtCn2cmhoE1RQbRlfHMpfUdEXf3P5SlSKHmDiNexSXhJlRzvpsG01iate9Uz8zWRqVu9ZlpviXd+SBuz+HOc8aByVXafsXh6XKmfZqPYwz4OaZ4nqKCKGurkPcAjnvBsVa3QSq27z1Qrl2l28W1ZDx9OJ5A8</latexit><latexit sha1_base64="+azdXGcR5olXvw3Vf8p4CiDiImE=">ACxnicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl05VUtA+opSTaR2aJmEyqZQi+ANu9dPEP9C/8M6YglpEJyQ5c+49Z+be68eBSJTjvOashcWl5ZX8amFtfWNzq7i90iVDJeZ1EQyZbvJTwQIa8roQLeiX3Rn7Am/7wXMebYy4TEYXahLzsgbhKIvmKeIuhp3L7rFklN2zLngZuBErJVi4ovuEPERhSjMARQhEO4CGhpw0XDmLiOpgSJwkJE+e4R4G0KWVxyvCIHdJ3QLt2xoa0156JUTM6JaBXktLGAWkiypOE9Wm2iafGWbO/eU+Np7bhP5+5jUiVuGW2L90s8z/6nQtCn2cmhoE1RQbRlfHMpfUdEXf3P5SlSKHmDiNexSXhJlRzvpsG01iate9Uz8zWRqVu9ZlpviXd+SBuz+HOc8aByVXafsXh6XKmfZqPYwz4OaZ4nqKCKGurkPcAjnvBsVa3QSq27z1Qrl2l28W1ZDx9OJ5A8</latexit>

σ

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δ(v − vi) → δσ(v − vi)

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∝ e− (v−vi)2

2σ2

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ρ(v)

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slide-6
SLIDE 6

Slow losses in a integrable gas (1D hard spheres or hard rods) L

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imagine that some of the particles are lost, at a rate

d dtρ(v) = −ΓF[ρ](v)

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The distribution of velocities evolves according to in this toy model. More generally, in this talk we will have:

ρ(v)

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v

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t = 0

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t4

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t3

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t2

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t1

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d dtρ(v) = −Γρ(v)

<latexit sha1_base64="ln2EHND+BDxuNdoEJWs7UYlAJ4Y=">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</latexit><latexit sha1_base64="ln2EHND+BDxuNdoEJWs7UYlAJ4Y=">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</latexit><latexit sha1_base64="ln2EHND+BDxuNdoEJWs7UYlAJ4Y=">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</latexit><latexit sha1_base64="ln2EHND+BDxuNdoEJWs7UYlAJ4Y=">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</latexit>

and the problem will be to determine the functional .

F[ρ]

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Γ ⌧ 1/τrelax

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slide-7
SLIDE 7

7

  • body losses: when atoms are at the same position, they can escape the system

K

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K

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x

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K

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atoms at the same position can escape the system

L

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Losses in the 1D Bose gas

(~ = m = 1)

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contact repulsion kinetic term

The 1D Bose gas is described by the Hamiltonian where

H = Z L ψ† ✓ −1 2∂2

x + g

2ψ†ψ ◆ ψ dx

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[ψ(x), ψ†(y)] = δ(x − y)

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This is described by the Lindblad equation for the density matrix: removes K atoms unitary evolution enforces tr ˆ

ρ = 1

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K-body loss rate

dˆ ρ dt = −i[H, ˆ ρ] + G Z ✓ ψK ˆ ρψ†K − 1 2{ψ†KψK, ˆ ρ} ◆ dx

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[G] = [lengthK−1/time]

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typical loss rate: with atom density

Γ = GnK−1

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n = ⌦ ψ†ψ ↵

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slide-8
SLIDE 8

The eigenstates of the Hamiltonian are superpositions of products of plane waves labeled by the rapidities

8

|{v1, v2, v3, . . . , vN}i

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i.e., for each eigenstate the wave function is of the form [Lieb, Liniger, Phys. Rev. 130 (1963)] where is a known phase. These eigenstates are called ‘Bethe states’. The rapidities can be thought of the asymptotic velocities, when one lets the system expand in 1D:

eiϕσ({vj})

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Rapidities in the 1D Bose gas

x

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L

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h0| ψ(x1) . . . ψ(xN) |{v1, . . . , vN}i = X

  • perm. σ

eiϕσ({vj})eivσ(1)x1+ivσ(2)x2+···+ivσ(N)xN

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x

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x

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|{v1, v2, . . . , vN}i

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v1

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v2

<latexit sha1_base64="d5L9Ua720Q8DlAfVSMs0ZuscCeg=">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</latexit><latexit sha1_base64="d5L9Ua720Q8DlAfVSMs0ZuscCeg=">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</latexit><latexit sha1_base64="d5L9Ua720Q8DlAfVSMs0ZuscCeg=">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</latexit><latexit sha1_base64="d5L9Ua720Q8DlAfVSMs0ZuscCeg=">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</latexit>

vN

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Time v1 < v2 < · · · < vN

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collisions during expansion no collision

ρ(v) = 1 L

N

X

j=1

δ(v − vj)

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The rapidity distribution is:

slide-9
SLIDE 9

9

Rapidities in the 1D Bose gas

The rapidities (i.e. asymptotic velocities) can be measured by letting the gas expand in 1D [Rigol-Muramatsu, PRL 94, 2005; Minguzzi-

Gangardt, PRL 94, 2005; Jukic-Pezer-Gasenzer-Buljan, PRA 78, 2008; Bolech-Heidrich-Meisner-Langer-McCulloch- Orso-Rigol, PRL 109, 2012; Bolech-Heidrich-Meisner- Langer-McCulloch-Orso-Rigol, J.o. Physics: Conference Series 414 2013, Campbell-Gangardt-Kheruntsyan, PRL 114, 2015; Caux-Doyon-JD-Konik-Yoshimura, SciPost 6, 2019, …]

x

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x

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v1

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v2

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vN

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Time

collisions during expansion no collision

experimental measurement of rapidity distribution

ρ(v) = 1 L

N

X

i=1

δ(v − vi)

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slide-10
SLIDE 10

10

From Lindblad to the rapidity distribution

The charges that would be conserved under unitary evolution (i.e. ) are no longer conserved under Lindblad evolution. Assuming losses are slow compared to relaxation, i.e. after each loss event the density matrix quickly relaxes towards a density matrix such that then the expectation value evolves as follows: In particular, we can apply this to the rapidity distribution itself (up to some smoothening) where is a smoothed Dirac delta function (e.g. a Gaussian of width ). This gives the evolution of the rapidity distribution:

[Q, H] = 0

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Q

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hQi = tr[ˆ ρQ]

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1 L d hQi dt = G ⌦ [ψ†K, Q]ψK↵

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Qσ(v) |{v1, . . . , vN}i =

N

X

j=1

δσ(v vj) |{v1, . . . , vN}i

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σ

<latexit sha1_base64="eDOTd40Hs6yYnbwKE0mQE+xkCW4=">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</latexit><latexit sha1_base64="eDOTd40Hs6yYnbwKE0mQE+xkCW4=">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</latexit><latexit sha1_base64="eDOTd40Hs6yYnbwKE0mQE+xkCW4=">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</latexit><latexit sha1_base64="eDOTd40Hs6yYnbwKE0mQE+xkCW4=">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</latexit>

δσ

<latexit sha1_base64="2xXxyTXM2ijqeCYL+WEHTntFPwQ=">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</latexit><latexit sha1_base64="2xXxyTXM2ijqeCYL+WEHTntFPwQ=">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</latexit><latexit sha1_base64="2xXxyTXM2ijqeCYL+WEHTntFPwQ=">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</latexit><latexit sha1_base64="2xXxyTXM2ijqeCYL+WEHTntFPwQ=">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</latexit>

with

expectation value w.r.t macrostate parameterized by ρ(v)

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d dtρ(v) = −Γ F[ρ](v)

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Big challenge: evaluate the functional F

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local operator (exp. decaying tails)

F[ρ](v) = lim

σ→0+ n1−K ⌦

[Qσ(v), ψ†K]ψK↵

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[ˆ ρ, H] = [ˆ ρ, Q] = 0

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Γ ⌧ 1/τrelax

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slide-11
SLIDE 11

11

The functional : results

F[ρ]

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  • 1. Ideal Bose gas regime: when , the atoms behave like non-interacting bosons.

In that regime, the rapidities are nothing but the (non-interacting) atoms’ velocities. Then:

  • 2. Hard-core regime: when , an exact calculation is possible by mapping to free fermions:

1-body loss: (and for because two or more atoms cannot be at the same position). This formula shows that: — in general, the effect of losses is non-linear in the rapidity distribution — it is non-local in rapidity space, i.e. is a function of for all

F[ρ](v) = K K! ρ(v)

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K atoms lost in each loss event comes from (Wick’s theorem for bosons)

⌦ ψ†KψK↵ hψ†ψiK = K!

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e ⌧ mg2/~2

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energy per atom scattering energy repulsion energy

e gn, mg2/~2

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F[ρ] = 0

<latexit sha1_base64="kAD+jzS6TlIT/SZdUWFkXDMhxCs=">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</latexit><latexit sha1_base64="kAD+jzS6TlIT/SZdUWFkXDMhxCs=">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</latexit><latexit sha1_base64="kAD+jzS6TlIT/SZdUWFkXDMhxCs=">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</latexit><latexit sha1_base64="kAD+jzS6TlIT/SZdUWFkXDMhxCs=">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</latexit>

K ≥ 2

<latexit sha1_base64="CAGx1Rey/cLOEI2asKl248xTMw=">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</latexit><latexit sha1_base64="CAGx1Rey/cLOEI2asKl248xTMw=">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</latexit><latexit sha1_base64="CAGx1Rey/cLOEI2asKl248xTMw=">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</latexit><latexit sha1_base64="CAGx1Rey/cLOEI2asKl248xTMw=">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</latexit>

F[ρ](v)

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ρ(w)

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w

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F[ρ](v) = ρ(v) + 2n π Z ρ(v) − ρ(w) (v − w)2 dw − 2π ρ(v)2 − ✓ 1 π Z ρ(w)dw v − w ◆2!

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slide-12
SLIDE 12

12

The functional : results

F[ρ]

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  • 3. General case: Markov chain summation over Bethe states [J.-S. Caux and P. Calabrese, Physical Review A74,031605 (2006),

J.-S. Caux, P. Calabrese, and N. A. Slavnov, Journalof Statistical Mechanics: Theory and Experiment2007,P01008 (2007), J.-S. Caux, J. Math. Phys. 50, 095214 (2009), J.-S. Caux and R. M. Konik, Physical review letters109,175301 (2012), V. Alba, arXiv preprint arXiv:1507.06994 (2015), …]

The conditional probability of a post-loss state given the pre-loss state is:

[Piroli-Calabrese, J. Phys. A 48, 2015]

and is taken as the weight in the GGE parameterized by the rapidity distrib- ution . Here .

[Pozsgay, J. Stat. Mech. P11017 2011]

F[ρ](v) = X

|{vi}i

X

|{wj}i

p({vi})p({wj}|{vi}) @n gK({vi}) × 2 4

N

X

i=1

δσ(v − vi) −

NK

X

j=1

δσ(w − wi) 3 5 1 A

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sum over pre- loss states sum over post-loss states

p({wj}|{vi}) =

  • h{wj}| ψK |{vj}i
  • 2

h{vi}| ψ†KψK |{vi}i

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|{vi}i

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p({vi})

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ρ(v)

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gK({vi}) = h{vi}| ψ†KψK |{vi}i nK

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It works, but it’s computationally heavy…

slide-13
SLIDE 13

13

Summary

  • 1. It is important (experimentally relevant) to understand the effect of atom losses on the rapidity

distribution in the 1D Bose gas: Evaluating the functional is challenging.

  • 2. In arXiv:2006.03583, we have computed in the ideal Bose gas regime (trivial result) and

hard-core regime (non-trivial result), and implemented a numerical method based on sampling of Bethe states, which works for arbitrary parameters but is computationally heavy.

  • 3. Many open questions remain: what about the quasi-condensate regime ([Rauer-Grisins-Mazets-

Schweigler-Rohringer—Langen-Schmiedmayer, PRL 116, 2016; Grisins-Rauer-Langen-Schmiedmayer-Mazets, PRA 93, 2016; Bouchoule-Schemmer-Henkel, SciPost 5, 2018; Johnson, Szigeti, Schemmer, Bouchoule, PRA 96, 2017; Schemmer-Bouchoule, PRL121, 2018])?

Can one find good approximations to evaluate more efficiently? [Caux-Doyon-JD-Konik-Yoshimura,

SciPost 6, 2019; Mallayya-Rigol-de Roeck, PRX 9, 2019; Friedman-Gopalakrishnan-Vasseur, PRB 101, 2020; Bastianello-De Nardis- De Luca, arXiv:2003.01702; Durnin-Bhaseen-Doyon, arXiv:2004.11030; Lopez-Piqueres-Ware-Gopalakrishnan-Vasseur, arXiv: 2005.13546]

Analytical progress needed on form factors and their summation…

dρ(v)/dt = −ΓF[ρ](v)

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F[ρ]

<latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit>

F[ρ]

<latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">ACy3icjVHLSsNAFD2Nr1pfVZdugkVwVRIR7LIgiBuhgn1AWyRJp21omgmTiVCrS3/Arf6X+Af6F94Zp6AW0QlJzpx7zp259/pJFKbScV5z1sLi0vJKfrWwtr6xuVXc3mkPBMBqwc84qLleymLwpjVZSgj1koE8Z+xJr+6FTFmzdMpCGPr+QkYd2xN4jDfh4kqiWfdbuiCHvXhdLTtnRy54HrgElmFXjxRd0ANHgAxjMSQhCN4SOlpw4WDhLgupsQJQqGOM9yjQN6MVIwUHrEj+g5o1zZsTHuVM9XugE6J6BXktHFAHk46QVidZut4pjMr9rfcU51T3W1Cf9/kGhMrMST2L9M+V+fqkWij4quIaSaEs2o6gKTJdNdUTe3v1QlKUNCnMI9igvCgXbO+mxrT6prV731dPxNKxWr9oHRZnhXt6QBuz/HOQ8aR2XKbuXx6VqxYw6jz3s45DmeYIqzlFDXc/xEU94ti6s1Lq17j6lVs54dvFtWQ8f606SBA=</latexit>

F[ρ]

<latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit><latexit sha1_base64="Exuc3byPbsHVkN1+z4XmNM/9XkE=">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</latexit>

Thank you!

slide-14
SLIDE 14

14

Calculating : remarks

F[ρ]

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  • A. Using typicality of eigenstates (‘Generalized ETH’), one might replace the double sum by a single sum
  • ver post-loss states. For a given ‘representative eigenstate’ :

However, numerically, this does not work well: we find that, to evaluate numerically, it is better to sum also over pre-loss states. But, analytically, this would be the way to go.

  • B. Analytical progress on summation of form factors crucially needed!
  • C. What about the quasi-condensate regime, ? How to calculate there?

Connection to theoretical and experimental works in that regime? [Rauer-Grisins-Mazets-Schweigler-Rohringer—

Langen-Schmiedmayer, PRL 116, 2016; Grisins-Rauer-Langen-Schmiedmayer-Mazets, PRA 93, 2016; Bouchoule-Schemmer-Henkel, SciPost 5, 2018; Johnson, Szigeti, Schemmer, Bouchoule, PRA 96, 2017; Schemmer-Bouchoule, PRL121, 2018]

  • D. In recent related works, various approximations have been proposed to tackle effects of integrability
  • breaking. Is it possible to use one of these to describe atom losses?

[Caux-Doyon-JD-Konik-Yoshimura, SciPost 6, 2019; Mallayya-Rigol-de Roeck, PRX 9, 2019; Friedman-Gopalakrishnan-Vasseur, PRB 101, 2020; Bastianello-De Nardis-De Luca, arXiv:2003.01702; Durnin-Bhaseen-Doyon, arXiv:2004.11030; Lopez-Piqueres-Ware-Gopalakrishnan- Vasseur, arXiv:2005.13546]

This is not obvious. In particular, for losses, the sum over particle-hole excitations does not truncate.

  • E. In the hard-core limit, the equation turns out to be exactly solvable. A special

role is played by the (non-hermitian) charges: Why is that so?

sum over post- loss states

|{vi}i

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F[ρ]

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F[ρ](v) / X

|{wj}i

  • h{wj}| ψK |{vi}i
  • 2

2 4

N

X

i=1

δσ(v vi)

NK

X

j=1

δσ(w wj) 3 5

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energy per atom scattering energy repulsion energy

e ' gn mg2/~2

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F[ρ]

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dρ(v)/dt = −ΓF[ρ](v)

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Q(z; {vi}) =

N

X

i=1

1 z − vi , z ∈ C.

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slide-15
SLIDE 15

15

Inhomogeneous profiles: GHD with losses

For comparison with experimental data, it is also important to consider inhomogeneous settings. Thanks to the 2016 breakthrough of GHD, nowadays this is straightforward:

number of atoms N has dropped by 15%, compared to t=0. Losses cannot be neglected

∂tρ + ∂x[veffρ] − (∂xV )∂vρ = −GnK−1F[ρ]

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(Also [Caux-Doyon-JD-Konik-Yoshimura, SciPost 6, 2019; Friedman-

Gopalakrishnan-Vasseur, PRB 101, 2020; Bastianello-De Nardis-De Luca, arXiv:2003.01702; Durnin-Bhaseen-Doyon, arXiv:2004.11030; Lopez-Piqueres- Ware-Gopalakrishnan-Vasseur, arXiv:2005.13546] for similar equations for

  • ther problems -not losses- )

another good reason to calculate !

F[ρ]

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V (x)

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V (x)

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x

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slide-16
SLIDE 16

16

Rapidities in the 1D Bose gas

The rapidities (i.e. asymptotic velocities) can be measured by letting the gas expand in 1D [Rigol-Muramatsu, PRL 94, 2005; Minguzzi-

Gangardt, PRL 94, 2005; Jukic-Pezer-Gasenzer-Buljan, PRA 78, 2008; Bolech-Heidrich-Meisner-Langer-McCulloch- Orso-Rigol, PRL 109, 2012; Bolech-Heidrich-Meisner- Langer-McCulloch-Orso-Rigol, J.o. Physics: Conference Series 414 2013, Campbell-Gangardt-Kheruntsyan, PRL 114, 2015; Caux-Doyon-JD-Konik-Yoshimura, SciPost 6, 2019, …]

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x

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v1

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v2

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vN

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Time

collisions during expansion no collision

experimental measurement of rapidity distribution

ρ(v) = 1 L

N

X

i=1

δ(v − vi)

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To measure the effect of losses on the rapidity distribution,

  • ne imagine doing something like this:

t = 0

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t = t1

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t = t3 > t2

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t = t2 > t1

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x

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n(x)

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x

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n(x)

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x

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n(x)

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x

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n(x)

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initial profile

t = 0

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t = t1

<latexit sha1_base64="/vkuGIWGnZsqZjhtWTosDkqR/W4=">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</latexit><latexit sha1_base64="/vkuGIWGnZsqZjhtWTosDkqR/W4=">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</latexit><latexit sha1_base64="/vkuGIWGnZsqZjhtWTosDkqR/W4=">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</latexit><latexit sha1_base64="/vkuGIWGnZsqZjhtWTosDkqR/W4=">ACyHicjVHLSsNAFD2Nr1pfVZdugkVwFSZtoXUhFNyIqwr2AbWUJ3WoWkSkolSiht/wK1+mfgH+hfeGVPQRdEJSe6ce86Zufe6kS8Sydh7zlhZXVvfyG8WtrZ3dveK+wftJExj7e80A/jrusk3BcBb0khfd6NYu5MXZ93MmFynfueZyIMLiRs4j3p84ECPhOZKgljyXA3tQLDGrUmN2pWYyq2ZXy5UzCsrMrlfrpm0xvUrIVjMsvuEWQ4TwkGIKjgCSYh8OEnp6sMEQEdbHnLCYIqHzHI8okDYlFieGQ+iEvmPa9TI0oL3yTLTao1N8emNSmjghTUi8mGJ1mqnzqXZW6DLvufZUd5vR3828poRK3BH6l27B/K9O1SIxQl3XIKimSCOqOi9zSXVX1M3NH1VJcogIU/GQ8jHFnlYu+mxqTaJrV71dP5DMxWq9l7GTfGpbkDXkzRXB60y5bNLPu6WmqwbNR5HOEYpzTPGhq4RBMt8hZ4xgtejSsjMh6M2TfVyGWaQ/xaxtMXkumRFg=</latexit>

t = t2

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t = t3

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expansion

ρ(v)

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v

<latexit sha1_base64="DGMq+RkdWHFNwvf5caZxY0oYmCA=">ACxHicjVHLSsNAFD2Nr/qunQTLIKrkoigKykI4rIF+4BaJlOa+g0CZNJoRT9Abf6beIf6F94Z5yCWkQnJDlz7j1n5t4bpiLKlOe9FpyFxaXleLq2vrG5tZ2aWe3mSW5ZLzBEpHIdhkXEQxb6hICd5OJQ9GoeCtcHih460xl1mUxNdqkvLuKBjEUT9igSKqPr4tlb2KZ5Y7D3wLyrCrlpRecIMeEjDkGIEjhiIsECjpwMfHlLiupgSJwlFJs5xjzXS5pTFKSMgdkjfAe06lo1prz0zo2Z0iqBXktLFIWkSypOE9WmuiefGWbO/eU+Np7bhP6h9RoRq3BH7F+6WeZ/dboWhT7OTA0R1ZQaRlfHrEtuqJv7n6pSpFDSpzGPYpLwswoZ312jSYzteveBib+ZjI1q/fM5uZ417ekAfs/xzkPmscV36v49ZNy9dyOuoh9HOCI5nmKq5Q8N4P+IJz86lI5zMyT9TnYLV7OHbch4+AGUoj3c=</latexit><latexit sha1_base64="DGMq+RkdWHFNwvf5caZxY0oYmCA=">ACxHicjVHLSsNAFD2Nr/qunQTLIKrkoigKykI4rIF+4BaJlOa+g0CZNJoRT9Abf6beIf6F94Z5yCWkQnJDlz7j1n5t4bpiLKlOe9FpyFxaXleLq2vrG5tZ2aWe3mSW5ZLzBEpHIdhkXEQxb6hICd5OJQ9GoeCtcHih460xl1mUxNdqkvLuKBjEUT9igSKqPr4tlb2KZ5Y7D3wLyrCrlpRecIMeEjDkGIEjhiIsECjpwMfHlLiupgSJwlFJs5xjzXS5pTFKSMgdkjfAe06lo1prz0zo2Z0iqBXktLFIWkSypOE9WmuiefGWbO/eU+Np7bhP6h9RoRq3BH7F+6WeZ/dboWhT7OTA0R1ZQaRlfHrEtuqJv7n6pSpFDSpzGPYpLwswoZ312jSYzteveBib+ZjI1q/fM5uZ417ekAfs/xzkPmscV36v49ZNy9dyOuoh9HOCI5nmKq5Q8N4P+IJz86lI5zMyT9TnYLV7OHbch4+AGUoj3c=</latexit><latexit sha1_base64="DGMq+RkdWHFNwvf5caZxY0oYmCA=">ACxHicjVHLSsNAFD2Nr/qunQTLIKrkoigKykI4rIF+4BaJlOa+g0CZNJoRT9Abf6beIf6F94Z5yCWkQnJDlz7j1n5t4bpiLKlOe9FpyFxaXleLq2vrG5tZ2aWe3mSW5ZLzBEpHIdhkXEQxb6hICd5OJQ9GoeCtcHih460xl1mUxNdqkvLuKBjEUT9igSKqPr4tlb2KZ5Y7D3wLyrCrlpRecIMeEjDkGIEjhiIsECjpwMfHlLiupgSJwlFJs5xjzXS5pTFKSMgdkjfAe06lo1prz0zo2Z0iqBXktLFIWkSypOE9WmuiefGWbO/eU+Np7bhP6h9RoRq3BH7F+6WeZ/dboWhT7OTA0R1ZQaRlfHrEtuqJv7n6pSpFDSpzGPYpLwswoZ312jSYzteveBib+ZjI1q/fM5uZ417ekAfs/xzkPmscV36v49ZNy9dyOuoh9HOCI5nmKq5Q8N4P+IJz86lI5zMyT9TnYLV7OHbch4+AGUoj3c=</latexit><latexit sha1_base64="DGMq+RkdWHFNwvf5caZxY0oYmCA=">ACxHicjVHLSsNAFD2Nr/qunQTLIKrkoigKykI4rIF+4BaJlOa+g0CZNJoRT9Abf6beIf6F94Z5yCWkQnJDlz7j1n5t4bpiLKlOe9FpyFxaXleLq2vrG5tZ2aWe3mSW5ZLzBEpHIdhkXEQxb6hICd5OJQ9GoeCtcHih460xl1mUxNdqkvLuKBjEUT9igSKqPr4tlb2KZ5Y7D3wLyrCrlpRecIMeEjDkGIEjhiIsECjpwMfHlLiupgSJwlFJs5xjzXS5pTFKSMgdkjfAe06lo1prz0zo2Z0iqBXktLFIWkSypOE9WmuiefGWbO/eU+Np7bhP6h9RoRq3BH7F+6WeZ/dboWhT7OTA0R1ZQaRlfHrEtuqJv7n6pSpFDSpzGPYpLwswoZ312jSYzteveBib+ZjI1q/fM5uZ417ekAfs/xzkPmscV36v49ZNy9dyOuoh9HOCI5nmKq5Q8N4P+IJz86lI5zMyT9TnYLV7OHbch4+AGUoj3c=</latexit>